Codex Wiki
geometry-and-topology.bigb
= Geometry and topology
{wiki}
= Algebraic topology
{parent=Geometry and topology}
{wiki}
= Mapping cylinder
{parent=Algebraic topology}
{wiki}
The mapping cylinder of $f:A\to X$ is
$$
M_f=(A\times[0,1]\sqcup X)/((a,0)\sim f(a)).
$$
It deformation retracts onto the copy of $X$.
= Real projective space
{title2=$\mathbb{RP}^n$}
{parent=Algebraic topology}
{c}
{wiki}
Real projective space is the antipodal quotient $\mathbb{RP}^n=S^n/(x\sim-x)$.
= Mapping-cylinder model of punctured real projective three-space
{parent=Real projective space}
The complement of an open three-ball in $\mathbb{RP}^3$ is the mapping cylinder of the antipodal covering $S^2\to\mathbb{RP}^2$. It deformation retracts to $\mathbb{RP}^2$ and has boundary $S^2$.
= Integral homology of real projective three-space
{parent=Real projective space}
The integral homology groups are
$$
H_i(\mathbb{RP}^3;\mathbb Z)=
\begin{cases}
\mathbb Z,&i=0,3,\\
\mathbb Z/2,&i=1,\\
0,&\text{otherwise}.
\end{cases}
$$
= Double of punctured real projective three-space
{parent=Real projective space}
Doubling $\mathbb{RP}^3$ minus an open ball along its spherical boundary gives $\mathbb{RP}^3\mathbin\#\mathbb{RP}^3$. Its fundamental group is
$$
(\mathbb Z/2)*(\mathbb Z/2)\cong D_\infty,
$$
and its universal cover is $S^2\times\mathbb R$.
= Homology
{parent=Algebraic topology}
{wiki=Homology_(mathematics)}
Homology assigns abelian groups $H_i(X)$ to a space, measuring cycles modulo boundaries in each dimension.
= Exact sequence
{parent=Homology}
{wiki}
An exact sequence has the image of each map equal to the kernel of the next.
= Exact sequences
{synonym}
= Commutative diagram
{parent=Homology}
{wiki}
A commutative diagram is a diagram of objects and morphisms in which every two directed paths with the same endpoints define the same morphism.
= Chain complex
{title2=$(C_\bullet,d)$}
{parent=Homology}
{wiki}
A chain complex is a sequence of abelian groups or modules and homomorphisms $d_i:C_i\to C_{i-1}$ satisfying $d_{i-1}d_i=0$. Its homology is $H_i(C)=\ker d_i/\operatorname{im}d_{i+1}$.
= Chain cycle
{parent=Chain complex}
An $i$-chain $z$ is a cycle when $d_i z=0$.
= Chain boundary
{parent=Chain complex}
An $i$-chain is a boundary when it belongs to $\operatorname{im}d_{i+1}$.
= Chain boundaries
{synonym}
= Chain coefficient
{parent=Chain complex}
The coefficients of a chain are the scalars multiplying its basis simplices.
= Chain map
{parent=Chain complex}
{wiki}
A chain map $f:C\to C'$ is a family $f_i:C_i\to C'_i$ satisfying $d'_if_i=f_{i-1}d_i$.
= Induced map on homology
{title2=$f_*$}
{parent=Chain map}
A chain map sends cycles to cycles and boundaries to boundaries, and therefore induces $f_*:H_i(C)\to H_i(C')$ by $[x]\mapsto[f_i(x)]$.
= Chain homotopy
{parent=Chain map}
{wiki}
Chain maps $f,g:C\to C'$ are chain homotopic when there are maps $h_i:C_i\to C'_{i+1}$ such that
$$
f_i-g_i=d'_{i+1}h_i+h_{i-1}d_i.
$$
Chain-homotopic maps induce the same map on homology.
= Mapping cone
{title2=$M(f)$}
{parent=Chain complex}
{wiki=Mapping_cone_(homological_algebra)}
For a chain map $f:C\to C'$, one mapping-cone convention takes $M(f)_i=C_{i-1}\oplus C'_i$ with a differential combining $d,d'$ and $f$. Its short exact sequence with $C'$ and a shift of $C$ produces a long exact sequence in homology.
= Short exact sequence of chain complexes
{parent=Chain complex}
{wiki=Exact_sequence}
A degreewise short exact sequence $0\to A\to B\to C\to0$ of chain complexes induces a long exact sequence of homology groups through the connecting homomorphisms.
= Long exact sequence in homology
{parent=Short exact sequence of chain complexes}
{wiki=Exact_sequence#Long_exact_sequence}
The connecting map sends a homology class in the quotient complex to the class obtained by lifting a representative, applying the middle differential, and identifying the result in the subcomplex.
= Connecting homomorphism
{parent=Long exact sequence in homology}
{wiki=Connecting_homomorphism}
The connecting homomorphism is the degree-shifting map produced by the lift-and-boundary construction in a long exact sequence.
= Connecting homomorphisms
{synonym}
= Universal coefficient theorem for homology
{parent=Homology}
{wiki=Universal_coefficient_theorem}
For an abelian coefficient group $G$, there is a split short exact sequence
$$
0\to H_i(X;\mathbb Z)\otimes G
\to H_i(X;G)
\to\operatorname{Tor}(H_{i-1}(X;\mathbb Z),G)\to0.
$$
For $G=\mathbb Q$, the Tor term vanishes.
= Rational homology
{parent=Universal coefficient theorem for homology}
{wiki}
Rational homology satisfies
$$
H_i(X;\mathbb Q)\cong H_i(X;\mathbb Z)\otimes_\mathbb Z\mathbb Q.
$$
It records the free rank of integral homology and discards torsion.
= Simplicial homology
{parent=Homology}
{wiki}
Simplicial homology computes homology from the boundary maps between free abelian groups generated by oriented simplices.
= Euler characteristic
{title2=$\chi$}
{parent=Simplicial homology}
{c}
{wiki}
For a finite complex,
$$
\chi(X)=\sum_i(-1)^i\dim_\mathbb QH_i(X;\mathbb Q).
$$
= Euler-Poincare formula
{parent=Euler characteristic}
{c}
{wiki=Euler_characteristic\#Euler–Poincaré_formula}
If a finite simplicial complex has $f_i$ simplices of dimension $i$, then
$$
\chi(X)=\sum_i(-1)^if_i.
$$
Writing $C_i=Z_i\oplus$ a lift of $B_{i-1}$ and $Z_i=H_i\oplus B_i$ makes the boundary dimensions cancel in the alternating sum.
= Barycentric subdivision
{parent=Simplicial homology}
{wiki}
The vertices of the barycentric subdivision of a simplicial complex are its nonempty faces, and its simplices are strict chains of faces.
= Barycentric subdivision of a tetrahedron
{parent=Barycentric subdivision}
The barycentric subdivision of a tetrahedron has f-vector
$$
(f_0,f_1,f_2,f_3)=(15,50,60,24).
$$
Its two-skeleton therefore has Euler characteristic $15-50+60=25$ and homology $H_0\cong\mathbb Z$, $H_1=0$, and $H_2\cong\mathbb Z^{24}$.
= Local homology
{parent=Homology}
{wiki}
The local homology at $x\in X$ is $H_i(X,X\setminus\{x\})$. It is preserved by homeomorphisms and, in a simplicial complex, is computed from the reduced homology of the local link.
= Fixed barycentre of the barycentric tetrahedral two-skeleton
{parent=Local homology}
In the two-skeleton of a barycentrically subdivided tetrahedron, the vertex corresponding to the whole tetrahedron has link equal to the connected 1-skeleton of the subdivided boundary, with 14 vertices and 36 edges. Its local $H_2$ therefore has rank $36-14+1=23$. Every other point has smaller local rank, so every self-homeomorphism fixes this vertex.
= Homotopy
{parent=Algebraic topology}
{wiki}
A homotopy between maps $f,g:Y\to X$ is a continuous map $H:Y\times[0,1]\to X$ with $H(y,0)=f(y)$ and $H(y,1)=g(y)$.
= Null-homotopic map
{parent=Homotopy}
{wiki=Homotopy#Null-homotopy}
A map is null-homotopic when it is homotopic to a constant map.
= Extension-null-homotopy criterion for a sphere
{parent=Null-homotopic map}
A map $f:S^{n-1}\to X$ extends to $D^n$ exactly when it is null-homotopic. An extension contracts radially through the disc; conversely, a null-homotopy descends through the cone quotient $CS^{n-1}\cong D^n$.
= Contractible space
{parent=Homotopy}
{wiki=Contractible_space}
A space is contractible when its identity map is homotopic to a constant map. Every map into a contractible space is null-homotopic.
= Homotopy equivalence
{parent=Homotopy}
{wiki}
Spaces $X$ and $Y$ are homotopy equivalent when there are continuous maps
$$
f:X\to Y,\qquad g:Y\to X
$$
such that $g\circ f$ is homotopic to $\operatorname{id}_X$ and $f\circ g$ is homotopic to $\operatorname{id}_Y$.
= Retract
{parent=Homotopy}
{wiki=Retraction_(topology)}
A subspace $A\subseteq X$ is a retract of $X$ when there is a continuous map $r:X\to A$ whose restriction to $A$ is the identity.
= Retract of a contractible space
{parent=Retract}
Every retract of a contractible space is contractible. If $i:A\hookrightarrow X$ is the inclusion, $r:X\to A$ is a retraction, and $H$ contracts $X$, then
$$
(a,t)\longmapsto r(H(i(a),t))
$$
contracts $A$.
= Homotopy extension property
{parent=Homotopy}
{wiki}
A pair $(X,A)$ has the homotopy extension property when every homotopy on $A$ whose initial map extends to $X$ can itself be extended to a homotopy on $X$.
= Collapsing a contractible cofibration
{parent=Homotopy extension property}
If $A\subseteq X$ is contractible and $(X,A)$ has the homotopy extension property, the quotient map
$$
q:X\to X/A
$$
is a <homotopy equivalence>. Extend a contraction of $A$ to $X$; its endpoint is constant on $A$ and therefore factors through $q$ to supply a homotopy inverse.
= Based loop
{parent=Algebraic topology}
{wiki=Loop_(topology)}
A based loop in $(X,x_0)$ is a path $\alpha:[0,1]\to X$ whose two endpoints are $x_0$.
= Based homotopy
{parent=Based loop}
{wiki=Homotopy\#Homotopy_relative_to_a_subspace}
A based homotopy between based loops keeps both endpoints fixed at the base point throughout the deformation.
= Path reversal
{parent=Based loop}
The reverse of a path $\alpha$ is $\bar\alpha(t)=\alpha(1-t)$. Concatenating a path with its reverse is homotopic relative to endpoints to the constant path.
= Fundamental group
{title2=$\pi_1(X,x_0)$}
{parent=Algebraic topology}
{wiki}
The fundamental group $\pi_1(X,x_0)$ consists of based loops modulo based homotopy, with multiplication induced by concatenation.
= Well-defined loop concatenation
{parent=Fundamental group}
Concatenating two based homotopies proves that loop concatenation depends only on based-homotopy classes. Endpoint-fixing reparametrizations provide associativity and the identity laws on classes.
= Functoriality of the fundamental group
{parent=Fundamental group}
{wiki=Fundamental_group\#Functoriality}
A continuous based map $f:(X,x_0)\to(Y,y_0)$ induces a homomorphism $f_*:\pi_1(X,x_0)\to\pi_1(Y,y_0)$ by composition of loops with $f$.
= Seifert-van Kampen theorem
{parent=Fundamental group}
{c}
{wiki}
For a suitably path-connected open cover $X=U\cup V$, the fundamental group of $X$ is the pushout of the homomorphisms from $\pi_1(U\cap V)$ to $\pi_1(U)$ and $\pi_1(V)$.
= Amalgamated free product
{parent=Seifert-van Kampen theorem}
{wiki=Free_product\#Generalization:_Free_product_with_amalgamation}
Given homomorphisms $i_A:C\to A$ and $i_B:C\to B$, the amalgamated free product $A*_CB$ comes with homomorphisms from $A$ and $B$ that agree on $C$. It is universal with this property: every compatible pair of homomorphisms $A\to H$ and $B\to H$ factors uniquely through $A*_CB$.
= Generation of a fundamental group by two open sets
{parent=Seifert-van Kampen theorem}
If $X=U_1\cup U_2$, the sets $U_1,U_2$, and their intersection are path-connected, and the base point lies in the intersection, then $\pi_1(X)$ is generated by the images of $\pi_1(U_1)$ and $\pi_1(U_2)$. Subdivide each loop into arcs lying in cover members and join every transition point to the base point within the intersection; the inserted paths telescope.
= Fundamental group after attaching a Möbius band
{parent=Seifert-van Kampen theorem}
A Möbius band retracts to its core circle, and its boundary wraps twice around the core. Attaching it to $X$ along a loop representing $w\in\pi_1(X)$ therefore adjoins a generator $x$ with relation $x^2=w$.
= Two Möbius bands attached to a torus
{parent=Fundamental group after attaching a Möbius band}
Attach Möbius bands to a torus along the classes $ab$ and $a^2b^3$. If their core generators are $x,y$, the resulting presentation is
$$
\langle a,b,x,y\mid[a,b],\ x^2=ab,\ y^2=a^2b^3\rangle.
$$
Because the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of $\mathbb Z^2$, eliminating $a,b$ gives
$$
\langle x,y\mid[x^2,y^2]=1\rangle.
$$
This group maps onto $S_3$ by $x\mapsto(12)$ and $y\mapsto(23)$, so it is nonabelian.
= Fundamental group after attaching a 2-cell
{parent=Seifert-van Kampen theorem}
Attaching a disc to $X$ along a based loop $\alpha$ kills precisely the normal closure of its homotopy class:
$$
\pi_1(X\cup_\alpha D^2)\cong
\pi_1(X)/\langle\!\langle[\alpha]\rangle\!\rangle.
$$
= Presentation complex
{parent=Fundamental group after attaching a 2-cell}
{wiki=Presentation_complex}
A group presentation gives a two-dimensional cell complex with one vertex, one oriented loop for each generator, and one 2-cell attached along each relator. Its fundamental group is the presented group.
= Presentation complex for the symmetric group on three letters
{parent=Presentation complex}
Attach three 2-cells to $S^1\vee S^1$ along loops representing $a^2$, $b^3$, and $(ab)^2$. The resulting group is
$$
\langle a,b\mid a^2,b^3,(ab)^2\rangle\cong S_3.
$$
Indeed, $a\mapsto(12)$ and $b\mapsto(123)$ gives a surjection to $S_3$, while $aba=b^{-1}$ reduces every word to one of $b^i$ or $ab^i$, so the presented group has at most six elements.
= Fundamental group of a closed orientable surface
{title2=$\pi_1(\Sigma_g)$}
{parent=Fundamental group}
{wiki=Surface_group}
For the closed orientable surface of genus $g$,
$$
\pi_1(\Sigma_g)
=\left\langle a_1,b_1,\ldots,a_g,b_g
\mathrel{\Big|}
\prod_{i=1}^g[a_i,b_i]=1\right\rangle.
$$
In particular, the sphere has trivial fundamental group and the torus has fundamental group $\mathbb Z^2$.
= Covering space
{parent=Algebraic topology}
{wiki}
A covering map $p:\widetilde X\to X$ is locally a disjoint union of homeomorphisms onto the same evenly covered open neighbourhood.
= Universal cover
{parent=Covering space}
{wiki=Universal_cover}
A universal cover is a simply connected covering space. The universal cover of a cylinder is the plane, with the angular coordinate unwrapped to a real coordinate.
= Uniqueness of a universal covering space
{parent=Universal cover}
For a path-connected locally path-connected base, any two based universal covers lift to one another. The two composites and the respective identity maps are based lifts of the same covering maps, so uniqueness of lifts makes the composites identities. The lift is therefore a homeomorphism.
= Extension criterion into a space with contractible universal cover
{parent=Universal cover}
Let $X$ have contractible universal cover, let $K$ be a path-connected simplicial complex, and let $i:K^1\hookrightarrow K$. A map $f:|K^1|\to X$ extends to $|K|$ exactly when its induced fundamental-group map factors through $i_*$. The factorization kills every 2-simplex boundary; all higher-dimensional boundary maps lift to the contractible universal cover and are null-homotopic.
= Classification of connected covering spaces
{parent=Covering space}
{wiki=Covering_space\#Lifting_properties}
Under the usual local hypotheses, based connected coverings of $X$ correspond to subgroups of $\pi_1(X)$; changing the point above the base point conjugates the subgroup. Normal subgroups correspond to regular coverings.
= Degree of a connected covering
{parent=Classification of connected covering spaces}
For a connected covering corresponding to $H\leq\pi_1(X)$, every fibre has cardinality $[\pi_1(X):H]$.
= Fibre bijection by path lifting
{parent=Degree of a connected covering}
If the base of a covering is path-connected, a path from $x_0$ to $x_1$ gives a bijection
$$
p^{-1}(x_0)\longrightarrow p^{-1}(x_1)
$$
by sending each initial lift point to the endpoint of its unique lifted path. Lifting the reversed path gives the inverse bijection.
= Cell complex of a covering from a coset graph
{parent=Classification of connected covering spaces}
For a presentation complex, the covering associated with a subgroup has one vertex for each coset. A generator gives directed edges according to its coset action, and every relator has one lifted 2-cell beginning at each vertex.
= Covering-space subgroup and deck-group quotient
{parent=Classification of connected covering spaces}
For a connected regular covering corresponding to $H\leq\pi_1(X,x_0)$, the subgroup $H$ is normal and
$$
\pi_1(X,x_0)/H\cong\operatorname{Deck}(\widetilde X/X).
$$
The quotient acts on a fibre by endpoints of lifted loops, and regularity makes this action transitive with kernel $H$.
= Normal covering map
{parent=Classification of connected covering spaces}
{wiki=Covering_space#Regular_(Galois)_covering}
A connected covering is normal, or regular, when its deck transformations act transitively on each fibre. Under the subgroup classification, this is equivalent to the associated subgroup of the fundamental group being normal.
= Universal covering map is normal
{parent=Normal covering map}
A universal covering has simply connected total space, so its associated subgroup of the base fundamental group is trivial. The trivial subgroup is normal; hence every universal covering map is a <normal covering map>.
= Forced normality of finite connected covers of orientable surfaces
{parent=Normal covering map}
For a connected degree-$n$ cover of a closed orientable surface $\Sigma_g$, normality is forced when $n=1$, when $n=2$, or when $g=1$. For $g=0$, existence itself forces $n=1$. For every $g\geq2$ and $n\geq3$, nonnormal connected degree-$n$ covers exist.
= Nonnormal finite cover of a higher-genus orientable surface
{parent=Forced normality of finite connected covers of orientable surfaces}
For $g\geq2$ and $n\geq3$, send
$$
a_1\mapsto\sigma,quad b_1\mapsto\tau,quad
a_2\mapsto\tau,quad b_2\mapsto\sigma,
$$
where $\sigma=(1\,2\,\cdots\,n)$ and $\tau=(1\,2)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.
= Path lifting theorem
{parent=Covering space}
{wiki=Covering_space\#Lifting_properties}
A path in the base of a covering has a unique lift after its initial point in the fibre is chosen. Subdivide its compact parameter interval so that each segment lies in an evenly covered set, then use the inverse sheet maps successively.
= Lifting criterion for a covering space
{parent=Covering space}
For a covering $p:(\widetilde X,\widetilde x_0)\to(X,x_0)$ and a based map $f:(Y,y_0)\to(X,x_0)$ from a path-connected locally path-connected space, a based lift $\widetilde f$ exists exactly when
$$
f_*\pi_1(Y,y_0)
\subseteq
p_*\pi_1(\widetilde X,\widetilde x_0).
$$
= Covering space action
{parent=Covering space}
{wiki=Covering_space\#Covering_action}
An action of $G$ on $X$ is a covering space action when every point has a neighbourhood $U$ disjoint from $gU$ for every nonidentity $g$. The quotient map is then locally a covering map, although without stronger properness assumptions its quotient need not be Hausdorff.
= Non-Hausdorff orbit space of a covering action
{parent=Covering space action}
On $\mathbb R^2\setminus\{0\}$, the powers of $T(x,y)=(2x,y/2)$ act freely with locally disjoint translates. Nevertheless $(2^{-n},1)\to(0,1)$ while $T^n(2^{-n},1)\to(1,0)$, so the two distinct limiting orbits cannot be separated in the quotient.
= Simply connected non-Hausdorff orbit-space example
{parent=Non-Hausdorff orbit space of a covering action}
Lifting the hyperbolic-scaling action to the universal cover of $\mathbb C^*$ preserves the two inseparable limiting orbits. The covering surface is biholomorphic to $\mathbb C$, so a simply connected Riemann surface can still have a non-Hausdorff quotient by a covering space action of homeomorphisms.
= Topological group
{parent=Algebraic topology}
{wiki}
A topological group is a group whose multiplication and inversion maps are continuous.
= Fundamental group of a topological group
{parent=Topological group}
For loops based at the identity of a topological group, pointwise multiplication and concatenation obey the interchange law and have the same identity. The Eckmann-Hilton argument therefore makes the fundamental group abelian.
= Eckmann-Hilton argument
{parent=Fundamental group of a topological group}
{c}
{wiki=Eckmann%E2%80%93Hilton_argument}
If two unital operations on one set share their identity and satisfy $(a*b)\cdot(c*d)=(a\cdot c)*(b\cdot d)$, then the operations coincide and are commutative.
= Unitary group
{title2=$U(n)$}
{parent=Topological group}
{wiki=Unitary_group}
The unitary group $U(n)$ consists of complex matrices preserving the standard Hermitian inner product.
= Determinant detects the order of a unitary loop
{parent=Unitary group}
The determinant $U(n)\to S^1$ sends a unitary loop to a circle-valued loop. A nonzero determinant winding number proves that the original loop class has infinite order.
= Lefschetz number
{title2=$L(f)$}
{c}
{parent=Algebraic topology}
{wiki}
$L(f)=\sum_i(-1)^i\operatorname{tr}(f_*:H_i\to H_i)$, equivalently the alternating trace on simplicial chains.
= Lefschetz fixed-point theorem
{c}
{parent=Algebraic topology}
{wiki}
If $L(f)\ne0$, then $f$ has a fixed point. For a fixed-point-free map, sufficiently fine simplicial approximation has zero diagonal chain coefficients, hence zero Lefschetz number.
= Mayer-Vietoris theorem
{c}
{parent=Algebraic topology}
{wiki=Mayer–Vietoris_sequence}
Mayer-Vietoris gives a long exact homology sequence for a space decomposed into two suitable subspaces.
= Snake lemma
{c}
{parent=Algebraic topology}
{wiki=Snake_lemma}
A commutative diagram of modules with exact rows yields an exact sequence from the kernels and cokernels of its three vertical maps. Applied degree by degree to a short exact sequence of chain complexes, it produces the connecting homomorphisms in the associated long exact homology sequence.
= Simplicial complex
{parent=Algebraic topology}
{wiki}
A simplicial complex is a family of finite vertex sets closed under taking subsets.
= Simplex
{parent=Simplicial complex}
{wiki}
An $n$-simplex has $n+1$ affinely independent vertices.
= Face of a simplex
{parent=Simplex}
{wiki=Simplex#Faces}
A face is the simplex spanned by a subset of the vertices.
= Orientation
{parent=Simplicial complex}
{wiki=Orientation_(vector_space)}
An orientation of a simplex is an ordering of its vertices modulo even permutations.
= Simplicial pseudomanifold
{parent=Simplicial complex}
{wiki=Pseudomanifold}
A pure $n$-dimensional simplicial complex is a pseudomanifold when every codimension-one simplex belongs to exactly two top-dimensional simplices and the top simplices are connected through shared codimension-one faces.
= Fundamental class of an orientable simplicial pseudomanifold
{parent=Simplicial pseudomanifold}
Compatible orientations of all top-dimensional simplices make their signed sum a cycle. For a connected pseudomanifold this cycle generates top homology over the integers; if no compatible orientation exists, top integral homology is zero.
= Simplicial cone
{parent=Algebraic topology}
{wiki=Cone_(topology)}
The simplicial cone $v*K$ adjoins a new vertex $v$ to every simplex of $K$ and is contractible.
= Suspension of a topological space
{parent=Algebraic topology}
{wiki=Suspension_(topology)}
The suspension is the union of two cones on a space along their common base.
= Reduced homology of a suspension
{parent=Suspension of a topological space}
Suspension shifts reduced homology: $\widetilde H_n(\Sigma X)\cong\widetilde H_{n-1}(X)$.
= Union of cones along a common base
{parent=Suspension of a topological space}
The union of $m$ cones on a nonempty space along one common base is homotopy equivalent to a wedge of $m-1$ suspensions of that space.
= Algebraic geometry
{parent=Geometry and topology}
{wiki}
= Algebraic variety
{parent=Algebraic geometry}
{wiki}
An algebraic variety is a geometric space locally described by polynomial equations, together with its regular functions.
= Smooth algebraic curve
{parent=Algebraic variety}
{wiki=Algebraic_curve}
A smooth algebraic curve is a one-dimensional <algebraic variety> whose <local ring> at every point is regular.
= Affine algebraic set
{parent=Algebraic variety}
{wiki=Affine_variety}
An affine algebraic set is the common zero set $V(I)$ of an <ideal> in a polynomial ring over a field.
= Coordinate ring
{title2=$k[X]$}
{parent=Affine algebraic set}
{wiki}
For an affine algebraic set $X\subseteq\mathbb A_k^n$, its coordinate ring is
$$
k[X]=k[x_1,\ldots,x_n]/I(X).
$$
The set $X$ is irreducible exactly when $k[X]$ is an <integral domain>.
= Vanishing loci of ideal sums and intersections
{parent=Affine algebraic set}
For ideals $I,J\subseteq k[x_1,\ldots,x_n]$,
$$
V(I+J)=V(I)\cap V(J),
\qquad
V(I\cap J)=V(I)\cup V(J).
$$
The second equality also follows from $V(IJ)=V(I)\cup V(J)$ and
$$
IJ\subseteq I\cap J.
$$
= Irreducible components of two intersecting complex hyperbolas
{parent=Affine algebraic set}
The algebraic set
$$
V(x^2+y^2-1,x^2-z^2-1)\subseteq\mathbb A_{\mathbb C}^3
$$
has the two irreducible components
$$
V(y-iz,x^2-z^2-1)
\quad\text{and}\quad
V(y+iz,x^2-z^2-1).
$$
Each is irreducible because the linear change $u=x-z$, $v=x+z$ identifies its <coordinate ring> with
$$
\mathbb C[u,v]/(uv-1)\cong\mathbb C[u,u^{-1}],
$$
an <integral domain>.
= Function field of an algebraic variety
{parent=Algebraic variety}
{wiki=Function_field_of_an_algebraic_variety}
The function field $k(X)$ of an irreducible <algebraic variety> consists of its <rational function>[rational functions].
= Function field
{synonym}
= Dimension from the function field
{parent=Function field of an algebraic variety}
For an irreducible variety $X$ over a field $k$,
$$
\dim X=\operatorname{trdeg}_k k(X).
$$
Consequently birational irreducible varieties have the same dimension.
= Projective model of a finitely generated field
{parent=Function field of an algebraic variety}
If $K/k$ is a finitely generated field extension, write $K=\operatorname{Frac}(A)$ for a finitely generated integral $k$-algebra $A$. The projective closure of the affine variety $\operatorname{Spec}A$ is an irreducible projective variety with function field $K$.
= Morphism of algebraic varieties
{parent=Algebraic variety}
{wiki=Morphism_of_algebraic_varieties}
A morphism of algebraic varieties is a map given locally by regular functions. A projective-coordinate formula defines a morphism wherever its homogeneous coordinate functions do not vanish simultaneously.
= Fiber of a morphism
{parent=Morphism of algebraic varieties}
{wiki=Fiber_(mathematics)}
For a <morphism of algebraic varieties> $f:X\to Y$ and $y\in Y$, the fiber is the inverse-image variety $X_y=f^{-1}(y)$.
= Fibres of a morphism
{synonym}
= Isomorphism of algebraic varieties
{parent=Morphism of algebraic varieties}
An isomorphism of algebraic varieties is a <morphism of algebraic varieties>[morphism] with a morphic inverse. It preserves all properties intrinsic to the variety.
= Finite morphism
{parent=Morphism of algebraic varieties}
{wiki=Finite_morphism}
A finite morphism is a morphism whose inverse image of every affine open set is affine and whose corresponding coordinate ring is finite as a module over the base coordinate ring. A nonconstant morphism between smooth projective curves is finite, and its degree is the number of points in a generic fiber counted with multiplicity.
= Projective space
{title2=$\mathbb P^n$}
{parent=Algebraic variety}
{wiki=Projective_space}
Projective space $\mathbb P^n$ is the set of one-dimensional subspaces of a vector space of dimension $n+1$, written in homogeneous coordinates $[x_0:\cdots:x_n]$.
= Projective point
{parent=Projective space}
A projective point is an equivalence class $[x_0:\cdots:x_n]$ of nonzero coordinate vectors under multiplication by a nonzero scalar.
= Projective plane
{title2=$\mathbb P^2$}
{parent=Projective space}
{wiki}
The projective plane is the two-dimensional <projective space> $\mathbb P^2$.
= Projective linear transformation
{parent=Projective space}
{wiki=Projective_linear_group}
An invertible linear map of the underlying vector space induces a projective linear transformation of $\mathbb P^n$; scalar multiples induce the same transformation.
= Projective variety
{parent=Projective space}
{wiki}
A projective variety is a closed algebraic subvariety of some projective space.
= Projective dimension theorem
{parent=Projective variety}
{c}
{wiki}
If projective varieties $X,Y\subseteq\mathbb P^n$ have dimensions $r,s$ and $r+s\geq n$, then $X\cap Y$ is nonempty and every component has dimension at least $r+s-n$.
= Projective curve
{parent=Projective variety}
{wiki=Algebraic_curve}
A projective curve is a one-dimensional <projective variety>.
= Homogeneous coordinate ring
{parent=Projective variety}
{wiki=Homogeneous_coordinate_ring}
For $X=V_+(I)\subseteq\mathbb P^n$, the homogeneous coordinate ring is the <graded ring> $k[x_0,\ldots,x_n]/I$.
= Product of projective varieties
{parent=Projective variety}
{wiki=Segre_embedding}
The product of projective varieties is projective via the Segre embedding. Dimensions add under products, and the product of smooth varieties is smooth.
= Coordinate projection
{parent=Product of projective varieties}
The coordinate projections from a product send $(x,y)$ to $x$ and $y$, respectively. For products of <projective variety>[projective varieties], they are <morphism of algebraic varieties>[morphisms].
= Segre embedding
{parent=Product of projective varieties}
{c}
{wiki}
The Segre embedding sends
$$
([x_0:\cdots:x_m],[y_0:\cdots:y_n])
\longmapsto [x_iy_j]_{i,j}
$$
and realizes $\mathbb P^m\times\mathbb P^n$ as a <projective variety>.
= Smooth quadric surface
{parent=Product of projective varieties}
{wiki=Quadric_(algebraic_geometry)}
Over an <algebraically closed field> of <characteristic> other than two, every smooth quadric surface in $\mathbb P^3$ becomes
$$
V(x_0x_3-x_1x_2),
$$
after a <projective linear transformation>. The <Segre embedding> identifies it with $\mathbb P^1\times\mathbb P^1$.
= Rulings of a smooth quadric surface
{parent=Smooth quadric surface}
{wiki=Ruling_(geometry)}
Under $Q\cong\mathbb P^1\times\mathbb P^1$, the fibres of the two <coordinate projection>[coordinate projections] are the two rulings of the <smooth quadric surface>. Two distinct fibres in one ruling are disjoint <projective line>[projective lines], while one fibre from each ruling meets in one point.
= Disjoint curves on a smooth quadric surface
{parent=Rulings of a smooth quadric surface}
For distinct $p,q\in\mathbb P^1$, the curves $\mathbb P^1\times\{p\}$ and $\mathbb P^1\times\{q\}$ are disjoint, smooth, and projective.
= Smooth projective curve
{parent=Projective variety}
{wiki=Algebraic_curve}
A smooth projective curve is a nonsingular projective variety of dimension one.
= Local ring of a smooth algebraic curve
{parent=Smooth projective curve}
At every point of a <smooth algebraic curve>, the <local ring> is a <discrete valuation ring>. Its valuation measures the <order of vanishing> of a <rational function> at that point.
= Extension of a rational map from a smooth projective curve
{parent=Smooth projective curve}
Every <rational map of projective varieties>[rational map] from a <smooth projective curve> to a <projective variety> extends uniquely to a <morphism of algebraic varieties>[morphism]. At a missing point, the <discrete valuation ring> of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.
= Failure of rational-map extension on a singular curve
{parent=Extension of a rational map from a smooth projective curve}
Let $\nu:\widetilde C\to C$ be the <normalization of a nodal curve>. Its rational inverse $C\dashrightarrow\widetilde C$ cannot extend over the node: the two branches give two distinct points of $\widetilde C$, whereas a morphism can assign only one image to the node.
= Irreducible topological space
{parent=Algebraic geometry}
{wiki=Irreducible_component}
A nonempty topological space is irreducible when it is not the union of two proper closed subsets. Equivalently, every two nonempty open subsets intersect.
= Irreducible component
{parent=Irreducible topological space}
{wiki}
An irreducible component is a maximal irreducible closed subset. Every algebraic variety is a finite union of its irreducible components.
= Noetherian Zariski topology
{parent=Irreducible topological space}
The Zariski topology on affine space is Noetherian because polynomial rings are Noetherian. Every closed subset is a finite union of irreducible closed subsets: a minimal counterexample would split into two smaller closed subsets, each already having such a decomposition.
= Zariski-closed set
{parent=Algebraic geometry}
{wiki=Zariski_topology}
A Zariski-closed set is a common zero set of <polynomials>. Arbitrary intersections and finite unions of such sets are again Zariski closed.
= Zariski-open set
{parent=Algebraic geometry}
{wiki=Zariski_topology}
A Zariski-open set is the complement of a <Zariski-closed set>. A <distinguished open set>[distinguished affine open] has the form $D(f)=\{x:f(x)\ne0\}$.
= Distinguished open set
{parent=Zariski-open set}
For a polynomial $f$, the distinguished open set $D(f)$ is the locus on which $f$ does not vanish.
= Hilbert Nullstellensatz
{parent=Algebraic geometry}
{c}
{wiki}
For an <algebraically closed field> $k$ and an <ideal> $I\subseteq k[x_1,\ldots,x_n]$, the strong Hilbert Nullstellensatz is
$$
I(V(I))=\sqrt I.
$$
Its weak form says that an ideal with empty <affine algebraic set>[affine zero set] is the unit ideal. Equivalently, if finitely many <polynomials> have no common zero, a polynomial combination of them equals one.
= Projective Nullstellensatz
{parent=Hilbert Nullstellensatz}
{c}
{wiki=Hilbert%27s_Nullstellensatz\#Projective_version}
Let $I\subseteq k[x_0,\ldots,x_n]$ be a <homogeneous ideal> over an <algebraically closed field>, and let
$$
\mathfrak m=(x_0,\ldots,x_n)
$$
be the irrelevant ideal. Then $V_+(I)=\varnothing$ exactly when either $I=(1)$ or $\sqrt I=\mathfrak m$. For a proper homogeneous ideal this is also equivalent to $\mathfrak m^N\subseteq I$ for some $N$.
= Irrelevant ideal of projective space
{title2=$S_+$}
{parent=Projective Nullstellensatz}
{wiki=Irrelevant_ideal}
In the standard graded <homogeneous coordinate ring> $S=k[x_0,\ldots,x_n]$ of $\mathbb P^n$, the irrelevant ideal is
$$
S_+=(x_0,\ldots,x_n).
$$
It vanishes only at the origin of the corresponding <affine cone>, which does not represent a <projective point>.
= Quasi-compactness of an affine variety
{parent=Hilbert Nullstellensatz}
Every open cover of an affine variety has a finite subcover. Refine it by distinguished opens $D(f_i)$. Their complements have no common point, so the Nullstellensatz writes $1$ as a finite combination of the $f_i$; the corresponding finite family of distinguished opens covers.
= Zariski density of the complex exponential graph
{parent=Algebraic geometry}
The graph $\{(x,e^x):x\in\mathbb C\}$ is Zariski dense in $\mathbb A^2_\mathbb C$. If $\sum_jp_j(x)e^{jx}=0$, repeated application of $D-m$ removes the largest exponential term while acting injectively on the others, proving inductively that every $p_j$ vanishes.
= Degree of a projective curve
{parent=Algebraic geometry}
{wiki=Degree_of_an_algebraic_variety}
If the Hilbert polynomial of the homogeneous coordinate ring of a projective curve is
$$
P_C(m)=dm+c,
$$
then $\deg C=d$. Equivalently, a sufficiently general hyperplane section has scheme-theoretic length $d$.
= Bézout theorem
{parent=Degree of a projective curve}
{c}
{wiki=Bézout%27s_theorem}
Two <projective plane curve>[projective plane curves] of degrees $m$ and $n$ with no common irreducible component have total <intersection multiplicity> $mn$. In particular, two nonempty projective plane curves always intersect.
= Projective plane curve
{parent=Bézout theorem}
A projective plane curve is a <projective curve> embedded in the <projective plane>, usually given by one homogeneous equation.
= Intersection multiplicity
{parent=Bézout theorem}
{wiki}
Intersection multiplicity measures the local order of contact of two algebraic subvarieties. Transverse intersections have multiplicity one.
= Hilbert polynomial
{title2=$P_X(m)$}
{parent=Degree of a projective curve}
{c}
{wiki}
The Hilbert function of a finitely generated graded algebra agrees in all sufficiently large degrees with a unique polynomial. For a projective variety of dimension $r$, its leading term is $\deg(V)m^r/r!$.
= Generic hyperplane section of a projective curve
{parent=Degree of a projective curve}
For a general linear form $\ell$ not vanishing identically on a projective curve, the exact sequence
$$
0\to A_C(-1)\xrightarrow{\ell}A_C\to A_C/(\ell)\to0
$$
shows that the hyperplane-section length is $P_C(m)-P_C(m-1)=\deg C$.
= Degree under a linear projective embedding
{parent=Degree of a projective curve}
Adjoining projective coordinates that vanish identically does not change the homogeneous coordinate ring or Hilbert polynomial. A linear embedding of projective space therefore preserves the degree of an embedded variety.
= Embedding dependence of projective degree
{parent=Degree of a projective curve}
Projective degree belongs to an embedding rather than to the abstract variety. A line and a smooth conic in $\mathbb P^2$ are both abstractly $\mathbb P^1$ but have degrees one and two.
= Twisted cubic
{parent=Algebraic geometry}
{wiki}
The twisted cubic is the image of
$$
[s:t]\mapsto[s^3:s^2t:st^2:t^3]
$$
in $\mathbb P^3$. Its ideal is generated by the three $2\times2$ minors of
$$
\begin{pmatrix}x_0&x_1&x_2\\x_1&x_2&x_3\end{pmatrix},
$$
and it has degree three.
= Nondegenerate projective variety
{parent=Twisted cubic}
A projective variety is nondegenerate when it is contained in no hyperplane of its ambient projective space.
= Degree obstruction to the twisted cubic being a complete intersection
{parent=Twisted cubic}
The twisted cubic is a <nondegenerate projective variety>, so every hypersurface containing it has degree at least two. At least two hypersurfaces are needed to cut out a curve in $\mathbb P^3$, making their degree product at least four rather than the curve's degree three.
= Genus distinguishes equal-degree projective curves
{parent=Algebraic geometry}
A twisted cubic and a smooth plane cubic embedded in $\mathbb P^3$ both have degree three, but their genera are zero and one. Thus equal projective degree does not imply isomorphism.
= Zariski tangent space
{title2=$T_pX$}
{parent=Algebraic geometry}
{c}
{wiki}
For $P\in X\subseteq\mathbb A^n$, the Zariski tangent space is the common kernel at $P$ of the differentials of all polynomials in $I(X)$.
= Dimension from minimum tangent dimension
{parent=Zariski tangent space}
For an affine variety, its dimension is the minimum of $\dim T_{X,P}$ over its points. A point is smooth when its tangent dimension attains this minimum and singular otherwise.
= Smooth locus of a variety
{parent=Zariski tangent space}
{wiki=Singular_point_of_an_algebraic_variety}
The smooth locus consists of points $P$ for which $\dim T_{X,P}=\dim X$. For an irreducible variety over a perfect field it is a nonempty <Zariski-open set>[Zariski-open subset] and therefore dense.
= Singular point of an algebraic variety
{parent=Smooth locus of a variety}
{wiki=Singular_point_of_an_algebraic_variety}
A point $p$ of an irreducible variety $X$ is singular when $\dim T_pX>\dim X$. Equivalently, its local ring is not regular. The singular locus is the complement of the smooth locus.
= Density of the smooth locus
{parent=Smooth locus of a variety}
If $X\subseteq\mathbb A^n$ is irreducible of dimension $d$, choose a nonzero $(n-d)$-rowed minor of a Jacobian matrix at one point of minimum tangent dimension. Its nonvanishing locus is a nonempty <Zariski-open set> on which the Jacobian has rank $n-d$, so every point there is smooth. Irreducibility makes every nonempty <Zariski-open set> dense.
= Jacobian criterion
{parent=Density of the smooth locus}
{wiki=Jacobian_criterion}
For an affine variety over a perfect field, the tangent-space codimension at a point is the rank of the Jacobian matrix of defining equations. The smooth locus of a pure $d$-dimensional variety in $\mathbb A^N$ is where this rank is $N-d$.
= Singular locus of a product with a smooth variety
{parent=Zariski tangent space}
Since $T_{(x,y)}(X\times Y)=T_xX\oplus T_yY$, if $Y$ is smooth then
$$
\operatorname{Sing}(X\times Y)=\operatorname{Sing}(X)\times Y.
$$
In particular, dimensions add.
= Tangent spaces of a product of two nodal line pairs
{parent=Zariski tangent space}
For
$$
X=Z(x_1^2-x_2^2,x_3^2-x_4^2)\subset\mathbb A^4
$$
in characteristic other than two,
$$
T_PX=\{v:x_1v_1-x_2v_2=0,\ x_3v_3-x_4v_4=0\}.
$$
The variety has dimension two. Its tangent dimension is two when both coordinate pairs are nonzero, three when exactly one pair vanishes, and four at the origin.
= Tangent-space obstruction between two unions of three lines
{parent=Zariski tangent space}
The union of the three coordinate axes in $\mathbb A^3$ and a union of three distinct lines through the origin in $\mathbb A^2$ are not <isomorphism of algebraic varieties>[isomorphic]. Their origins are intrinsically the unique points common to all three <irreducible components>, but their <Zariski tangent space>[Zariski tangent spaces] there have dimensions three and two respectively.
= Krull dimension of an affine variety
{parent=Algebraic geometry}
{wiki=Krull_dimension}
The Krull dimension of an affine variety is the supremum of the lengths of strict chains of irreducible closed subsets, equivalently the Krull dimension of its coordinate ring. For an irreducible affine variety it equals the minimum Zariski tangent-space dimension.
= Affine hypersurface
{parent=Algebraic geometry}
{wiki=Algebraic_hypersurface}
The zero set of a nonconstant polynomial in $n$ affine variables has dimension $n-1$. Passing to the square-free part gives its radical principal ideal.
= Tangent hyperplane section
{parent=Affine hypersurface}
At a smooth point of a hypersurface, restriction to the tangent hyperplane has zero linear term. The natural hyperplane intersection therefore has tangent dimension one larger than its expected dimension and is singular at that point.
= Singular points of an irreducible affine plane cubic
{parent=Affine hypersurface}
An irreducible affine plane cubic has at most one singular point. If two existed, restriction of its cubic polynomial to their joining line would have a zero of multiplicity at least two at each point. A polynomial of degree at most three cannot have those four zeros unless it vanishes identically, making the line a component and contradicting irreducibility.
= Singular cylinder over a nodal curve
{parent=Affine hypersurface}
The affine surface
$$
V\bigl(y^2-x(x-1)^2\bigr)\subseteq\mathbb A^3_{x,y,z}
$$
is birational to $\mathbb A^2$ and has singular locus $\{(1,0,z):z\in k\}$, an irreducible affine line.
= Projective hypersurface
{parent=Algebraic geometry}
{wiki=Algebraic_hypersurface}
A projective hypersurface in $\mathbb P^n$ is the zero locus of one nonconstant homogeneous polynomial.
= Affine cone
{parent=Projective hypersurface}
{wiki}
For a projective algebraic set $X\subseteq\mathbb P^n$, its affine cone is the union in $\mathbb A^{n+1}$ of the lines represented by the points of $X$, together with the origin.
= Pencil of plane curves
{parent=Projective hypersurface}
{wiki=Pencil_(mathematics)}
Given homogeneous polynomials $F,G$ of the same degree, their pencil is the one-parameter family
$$
V(sF+tG)\subseteq\mathbb P^2,
\qquad [s:t]\in\mathbb P^1.
$$
Its total space is the <projective hypersurface> $V(sF+tG)\subseteq\mathbb P^2\times\mathbb P^1$, and projection to $[s:t]$ has these plane curves as its <fibres of a morphism>[fibres].
= Fermat cubic curve
{title2=$x^3+y^3+z^3=0$}
{parent=Pencil of plane curves}
{c}
{wiki=Fermat_curve}
The Fermat cubic $x^3+y^3+z^3=0$ is a <smooth projective curve>: its three first partial derivatives vanish simultaneously only at the forbidden zero vector. The <genus of a smooth plane curve> formula gives genus one.
= Cubic pencil with a triangular member
{parent=Pencil of plane curves}
The pencil generated by the <Fermat cubic curve> and $xyz=0$ contains both a smooth <genus one> curve and the union of the three coordinate lines. Its total space in $\mathbb P^2\times\mathbb P^1$ is irreducible because the two generating cubics have no common factor.
= Affine cone over a projective hypersurface
{parent=Projective hypersurface}
{wiki=Affine_cone}
If $X=V_+(F)\subseteq\mathbb P^n$, its affine cone is $Y=V(F)\subseteq\mathbb A^{n+1}$. If $X$ is smooth, every nonzero point of $Y$ is smooth; hence the vertex is the only possible singular point. The vertex is smooth for linear $F$ and singular whenever $\deg F\geq2$.
= Projective quadric cone with one singular vertex
{parent=Projective hypersurface}
Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadric
$$
V(X_0^2+\cdots+X_r^2)\subseteq\mathbb P^{r+1}
$$
is an irreducible $r$-dimensional cone for $r\geq2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots:0:1]$.
= Projective variety with a prescribed-dimensional singular locus
{parent=Projective quadric cone with one singular vertex}
For $n>k\geq0$, take an $(n-k)$-dimensional irreducible projective variety with one singular point and form its product with $\mathbb P^k$. The Segre embedding makes the product projective, and its singular locus is the singular point times $\mathbb P^k$, of dimension $k$. A quadric cone works in dimension at least two and a cuspidal cubic works in dimension one.
= Determinantal variety
{parent=Algebraic geometry}
{wiki}
A determinantal variety is cut out by minors imposing an upper bound on matrix rank.
= Smooth projective surface from overlapping rank-one coordinates
{parent=Determinantal variety}
The projective locus where
$$
\begin{pmatrix}y_0&y_1&y_2\\y_2&y_3&y_4\end{pmatrix}
$$
has rank one is cut out by its three $2\times2$ minors. It is a smooth surface: on each of the charts $y_0\ne0$, $y_1\ne0$, $y_3\ne0$, and $y_4\ne0$, the equations eliminate three coordinates and leave two free affine coordinates; these charts cover the locus.
= Rank-one determinantal variety
{parent=Determinantal variety}
The variety of $m\times n$ matrices of rank at most one has dimension $m+n-1$. For $m,n\geq2$, its only singular point is the zero matrix.
= Birational variety
{parent=Algebraic geometry}
{wiki=Birational_geometry}
Irreducible varieties are birational exactly when they have isomorphic function fields.
= Blowup of the affine plane at the origin
{parent=Birational variety}
{wiki=Blowing_up}
The blowup of $\mathbb A^2$ at the origin is
$$
\operatorname{Bl}_0\mathbb A^2
=\{((X,Y),[W:Z]):XZ=WY\}\subseteq\mathbb A^2\times\mathbb P^1.
$$
Its projection to $\mathbb A^2$ is an isomorphism away from the origin, while the fiber above the origin is the exceptional curve $\mathbb P^1$.
= Normalization of an algebraic curve
{parent=Algebraic geometry}
{wiki=Normalization_(algebraic_geometry)}
The normalization of an irreducible algebraic curve is a normal, hence smooth, curve with a finite birational morphism to the original curve.
= Geometric genus
{title2=$g(C)$}
{parent=Normalization of an algebraic curve}
{wiki=Geometric_genus}
The geometric genus of an irreducible curve is the genus of its smooth projective normalization.
= Genus one curve
{parent=Geometric genus}
{wiki=Genus_one_curve}
A genus one curve is a smooth projective curve of <geometric genus> one. Choosing a rational point makes it an elliptic curve.
= Genus one
{synonym}
= Normalization of a nodal curve
{parent=Normalization of an algebraic curve}
{wiki=Normalization_(algebraic_geometry)}
Normalization separates a node's two branches; a rational nodal cubic has normalization $\mathbb P^1$ and two points above its node.
= Normalization of y squared equals x times x minus one squared
{title2=$y^2=x(x-1)^2$}
{parent=Normalization of a nodal curve}
For the nodal affine cubic $y^2=x(x-1)^2$, the rational parameter
$$
t=\frac y{x-1}
$$
gives $x=t^2$ and $y=t(t^2-1)$. Its function field is $k(t)$, so its smooth projective normalization is $\mathbb P^1$ and has genus zero.
= Divisor on an algebraic curve
{parent=Algebraic geometry}
{wiki=Divisor_(algebraic_geometry)}
A divisor is a finite formal integer combination of closed points. For a divisor $D$,
$$
L(D)=\{f\in k(X)^\times:(f)+D\geq0\}\cup\{0\},
\qquad \ell(D)=\dim_kL(D).
$$
= Degree of a divisor
{title2=$\deg D$}
{parent=Divisor on an algebraic curve}
{wiki=Divisor_(algebraic_geometry)#Degree}
For $D=\sum_Pn_PP$ on a curve over an algebraically closed field, its degree is $\deg D=\sum_Pn_P$.
= Linear equivalence of divisors
{parent=Divisor on an algebraic curve}
{wiki=Divisor_(algebraic_geometry)#Linear_equivalence}
Two divisors $D$ and $E$ are linearly equivalent, written $D\sim E$, when $D-E$ is a <principal divisor on an algebraic curve>[principal divisor]. Multiplication by a rational function whose divisor is $D-E$ gives an isomorphism between their spaces of sections, so $\ell(D)=\ell(E)$.
= Principal divisor on an algebraic curve
{title2=$(f)$}
{parent=Divisor on an algebraic curve}
{wiki=Divisor_(algebraic_geometry)#Principal_divisors}
For a nonzero <rational function> $f$ on a <smooth projective curve> $C$, its principal divisor is
$$
(f)=\operatorname{div}(f)=\sum_{p\in C}\operatorname{ord}_p(f)p.
$$
It records the zeros of $f$ with positive multiplicity and its poles with negative multiplicity. Every principal divisor has <degree of a divisor>[degree] zero.
= Divisor class on the projective line
{parent=Principal divisor on an algebraic curve}
Every degree-zero divisor on the <projective line> is principal. Indeed, if $D=\sum_i a_ip_i$ and $L_i$ is a homogeneous linear form vanishing at $p_i$, then $\sum_i a_i=0$ makes
$$
f=\prod_iL_i^{a_i}
$$
a degree-zero rational function with $(f)=D$. Consequently, two divisors on $\mathbb P^1$ are linearly equivalent exactly when they have the same degree.
= Principal divisor with one simple zero and one simple pole
{parent=Principal divisor on an algebraic curve}
If distinct points $p,q$ on a <smooth projective curve> satisfy $(f)=p-q$, then $f:C\to\mathbb P^1$ is a <finite morphism> of degree one. Such a morphism between smooth projective curves is an <isomorphism of algebraic varieties>[isomorphism], so $C$ has <geometric genus>[genus] zero. Hence no divisor $p-q$ on a curve of positive genus is principal.
= Complete linear system of a divisor
{title2=$|D|$}
{parent=Divisor on an algebraic curve}
{wiki=Linear_system_of_divisors}
The complete linear system $|D|$ consists of the effective divisors linearly equivalent to $D$. A basis of $L(D)$ defines a rational map to $\mathbb P^{\ell(D)-1}$.
= Very ample divisor
{parent=Complete linear system of a divisor}
{wiki=Ample_line_bundle#Very_ample_line_bundles}
A divisor is very ample when its complete linear system defines a closed embedding into projective space.
= High-degree divisor is very ample on a smooth projective curve
{parent=Very ample divisor}
On a smooth projective curve of genus $g$, every divisor of degree at least $2g+1$ is very ample. Riemann--Roch shows that its sections separate distinct points and tangent directions by comparing $L(D)$ with $L(D-P-Q)$ and $L(D-2P)$.
= Riemann-Roch theorem
{parent=Divisor on an algebraic curve}
{c}
{wiki=Riemann–Roch_theorem}
For a smooth projective curve of genus $g$ and a canonical divisor $K$,
$$
\ell(D)-\ell(K-D)=\deg D+1-g.
$$
= Canonical divisor
{title2=$K_X$}
{parent=Riemann-Roch theorem}
{wiki}
A canonical divisor is the divisor of any nonzero rational differential. Its divisor class is independent of the differential and has degree $2g-2$.
= Canonical divisor of the projective line
{title2=$K_{\mathbb P^1}$}
{parent=Canonical divisor}
For the affine coordinate $t$ on $\mathbb P^1$, put $s=t^{-1}$ at infinity. Since $dt=-s^{-2}ds$,
$$
(dt)=-2\infty,
$$
so $K_{\mathbb P^1}\sim-2\infty$. More generally,
$$
\ell(d\infty)=\max(d+1,0),
$$
because $L(d\infty)$ consists of polynomials of degree at most $d$ when $d\geq0$ and only the zero function when $d<0$.
= Genus of a smooth plane curve
{parent=Canonical divisor}
{wiki=Genus–degree_formula}
A smooth plane curve of degree $d$ has genus
$$
g=\frac{(d-1)(d-2)}2.
$$
= Smooth plane quartic
{parent=Genus of a smooth plane curve}
{wiki=Quartic_plane_curve}
A smooth plane quartic is a smooth degree-four curve in $\mathbb P^2$ and has genus three.
= Canonical map
{parent=Canonical divisor}
{wiki}
For $g\geq2$, a basis of the space of regular differentials defines the canonical map $X\to\mathbb P^{g-1}$. Equivalently, after choosing a differential with divisor $K$, one may use a basis of $L(K)$.
= Gonality
{parent=Algebraic geometry}
{wiki=Gonality}
The gonality of a smooth projective curve is the least degree of a nonconstant morphism from the curve to the projective line.
= Hyperelliptic curve
{parent=Algebraic geometry}
{wiki}
A hyperelliptic curve of genus at least two admits a degree-two map to $\mathbb P^1$ and can be represented in characteristic other than two by an equation $y^2=f(x)$ with $f$ square-free.
= Even-degree hyperelliptic model
{parent=Hyperelliptic curve}
If $y^2=f(x)$ with $\deg f=2n$, the coordinates
$$
u=x^{-1},\qquad v=yx^{-n}
$$
give the second chart
$$
v^2=u^{2n}f(u^{-1}).
$$
There are two points $P_+,P_-$ above infinity.
= Compactification of y squared equals x to the eighth minus one
{parent=Even-degree hyperelliptic model}
The affine germ surface over the plane with the eight roots of unity removed is compactified by adding the eight simple branch points above those roots and two unbranched points above infinity. The degree-two map to the Riemann sphere has total ramification eight, so the <Riemann-Hurwitz formula> gives genus three.
= Canonical divisor of an even-degree hyperelliptic curve
{parent=Even-degree hyperelliptic model}
For the smooth curve $y^2=f(x)$ with square-free $f$ of degree $2n$, the differential $dx/y$ has divisor
$$
(n-2)(P_++P_-).
$$
Thus the canonical degree is $2n-4$, the genus is $n-1$, and $L(K)$ has basis $1,x,\ldots,x^{n-2}$.
= Canonical map of a hyperelliptic curve
{parent=Canonical divisor of an even-degree hyperelliptic curve}
The canonical map of an even-degree hyperelliptic curve is
$$
(x,y)\longmapsto[1:x:\cdots:x^{n-2}],
$$
so it identifies the two points $(x,y)$ and $(x,-y)$ of a general fibre and is not an embedding.
= Rational map of projective varieties
{parent=Algebraic geometry}
{wiki=Rational_mapping}
A rational map is a morphism on a dense open subset, considered up to agreement on a smaller dense open subset.
= Indeterminacy locus
{parent=Rational map of projective varieties}
{wiki=Indeterminacy_locus}
The indeterminacy locus is the set where a rational map has no regular local representative.
= Projective Cremona transformation
{parent=Rational map of projective varieties}
{wiki=Cremona_transformation}
A projective Cremona transformation is a birational self-map of projective space, regular on complementary dense open subsets together with its inverse.
= Nonsingular plane cubic
{parent=Algebraic geometry}
{wiki=Elliptic_curve}
A plane cubic is nonsingular when its homogeneous equation and all first partial derivatives have no common projective zero.
= Plane curve
{parent=Algebraic geometry}
{wiki}
A plane curve is the zero set of a polynomial in two variables, or a one-dimensional curve embedded in a plane.
= Canonical map of a smooth plane curve
{parent=Plane curve}
For a smooth plane curve of degree $d\geq4$, adjunction gives
$$
K_C\cong\mathcal O_C(d-3).
$$
Its canonical map is induced by the $(d-3)$rd Veronese map and is therefore an embedding.
= Semicubical parabola
{parent=Plane curve}
{wiki}
A semicubical parabola is a cusped cubic curve affinely equivalent to $y^2=x^3$.
= Ramification divisor
{title2=$R_f$}
{parent=Algebraic geometry}
{wiki=Ramification_(mathematics)}
For a finite morphism $f:X\to Y$ of smooth curves, the ramification divisor is
$$
R_f=\sum_{P\in X}(e_P-1)P.
$$
The <Riemann-Hurwitz formula> states $2g_X-2=(\deg f)(2g_Y-2)+\deg R_f$.
= Product surface without low-genus curves
{parent=Algebraic geometry}
If $C$ is a smooth projective curve of genus at least three, then the smooth projective surface $C\times C$ contains no curve of geometric genus below three. On the normalization of any curve in the product, at least one coordinate projection to $C$ is nonconstant, and the <Riemann-Hurwitz formula> cannot decrease genus.
= Projection of a plane curve from an exterior point
{parent=Algebraic geometry}
After coordinates place $p=[0:0:1]$, projection away from $p$ is $[x:y:z]\mapsto[x:y]$. Its restriction to a plane curve avoiding $p$ is a morphism to $\mathbb P^1$ whose degree equals the degree of the curve.
= Differential geometry
{parent=Geometry and topology}
{wiki}
= Immersion
{parent=Differential geometry}
{wiki=Immersion_(mathematics)}
An immersion is a smooth map whose differential is injective at every point. A parametrized surface in $\mathbb R^3$ is immersed when its two coordinate tangent vectors are linearly independent.
= Isothermal coordinates
{parent=Immersion}
{wiki}
Coordinates $(u,v)$ on an immersed surface are isothermal when the first fundamental form is $\lambda^2(du^2+dv^2)$ for a positive function $\lambda$.
= Laplace-Beltrami operator
{title2=$\Delta_g$}
{parent=Differential geometry}
{c}
{wiki}
The Laplace-Beltrami operator is the intrinsic Laplacian determined by a Riemannian metric. In local coordinates it is
$$
\Delta_gf=|g|^{-1/2}\partial_i\left(|g|^{1/2}g^{ij}\partial_jf\right).
$$
= Smooth surface
{parent=Differential geometry}
{wiki=Surface_(topology)}
A smooth surface in $\mathbb R^3$ is a two-dimensional smooth embedded submanifold. Locally it is parametrized by two coordinates with linearly independent tangent vectors.
= Smooth manifold
{parent=Differential geometry}
{wiki=Manifold}
A smooth $k$-manifold is a Hausdorff second-countable space locally homeomorphic to $\mathbb R^k$, equipped with smoothly compatible coordinate charts.
= Abstract smooth surface
{parent=Smooth manifold}
An abstract smooth surface is a two-dimensional <smooth manifold>. Thus it is a Hausdorff second-countable space with an atlas of plane-valued coordinate charts whose transition maps are smooth.
= Smooth quotient by a free finite group action
{parent=Abstract smooth surface}
A finite group acting freely by diffeomorphisms on a <smooth manifold> has a smooth quotient. Choose each coordinate neighbourhood disjoint from all its nontrivial translates and transport its chart through the quotient map. The resulting transition maps are restrictions of the original transition maps composed with group elements.
= Smooth map between manifolds
{parent=Smooth manifold}
{wiki=Smoothness}
A map between smooth manifolds is smooth when its expression in every pair of coordinate charts is a <smooth function>.
= Orientable smooth manifold
{parent=Smooth manifold}
{wiki=Orientability}
A smooth manifold is orientable when it admits an atlas whose transition maps have positive Jacobian determinant wherever they are defined.
= Orientation-reversing diffeomorphism
{parent=Orientable smooth manifold}
{wiki=Orientation_(vector_space)}
An orientation-reversing diffeomorphism reverses the chosen orientation. In oriented local coordinates its derivative has negative determinant.
= Riemannian metric
{title2=$g$}
{parent=Differential geometry}
{c}
{wiki}
A Riemannian metric assigns a positive-definite inner product to every tangent space, varying smoothly from point to point.
= Tangent vector
{parent=Differential geometry}
{wiki}
A tangent vector to a smooth curve is the <derivative> of a parametrization of that curve. Tangent vectors to a surface form its tangent plane.
= Tangent space
{title2=$T_pM$}
{parent=Tangent vector}
{wiki}
The tangent space $T_pM$ is the <vector space> of tangent vectors to a <smooth manifold> $M$ at $p$. For a regular parametrized surface $X(u,v)$, it is spanned by $X_u$ and $X_v$.
= Unit tangent vector
{title2=$T$}
{parent=Tangent vector}
{wiki=Frenet%E2%80%93Serret_formulas}
The unit tangent vector of a regular parametrized curve $\mathbf r(t)$ is
$$
T(t)=\frac{\mathbf r'(t)}{|\mathbf r'(t)|}.
$$
= Velocity vector
{parent=Tangent vector}
{wiki=Velocity}
The velocity vector of a parametrized curve $\mathbf r(t)$ is its tangent vector $\mathbf r'(t)$.
= Normal vector
{parent=Differential geometry}
{wiki}
A normal vector is orthogonal to every <tangent vector> in the tangent space.
= Unit normal
{parent=Normal vector}
{wiki=Normal_(geometry)}
A unit normal is a <normal vector> of norm one.
= Orientation of a surface
{parent=Differential geometry}
{wiki}
An orientation of a regular surface is a continuous choice of <unit normal>. Reversing orientation replaces $N$ by $-N$.
= Boundary orientation
{parent=Orientation of a surface}
{wiki=Stokes%27_theorem#Topological_preliminaries}
An orientation of a surface induces the boundary direction used in <Stokes theorem>. Walking in the positive boundary direction keeps an upward-oriented surface on the left.
= First fundamental form
{title2=$I$}
{parent=Differential geometry}
{wiki}
The first fundamental form of a surface $S\subseteq\mathbb R^3$ is the restriction of the ambient <inner product> to each <tangent space> $T_pS$. For a parametrized surface $X(u,v)$, it is
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,
\qquad
E=X_u^2,\quad F=X_u\cdot X_v,\quad G=X_v^2.
$$
= Local isometry
{parent=First fundamental form}
{wiki}
A local isometry is a smooth map whose differential preserves the inner product on every tangent space. It preserves intrinsic quantities such as <Gaussian curvature>.
= Geodesic preservation by a local isometry
{parent=Local isometry}
A local isometry intertwines the Levi--Civita covariant derivatives. It therefore sends every affinely parametrized geodesic to an affinely parametrized geodesic.
= Geodesic-preserving homothety that is not a local isometry
{parent=Geodesic preservation by a local isometry}
A nonunit Euclidean dilation sends straight-line geodesics to straight-line geodesics but scales their tangent inner products. Preservation of geodesics alone therefore does not imply local isometry.
= Riemannian isometry
{parent=Local isometry}
{wiki=Isometry_(Riemannian_geometry)}
A Riemannian isometry is a diffeomorphism that preserves the Riemannian metric, and hence lengths, areas, and intrinsic curvature.
= Local isometry from a circular cone to the plane
{parent=Local isometry}
For the cone
$$
X(r,\theta)=(r\cos\theta,r\sin\theta,\sqrt a\,r),
\qquad r>0,
$$
the <first fundamental form> is
$$
(1+a)\,dr^2+r^2\,d\theta^2.
$$
The local change of <polar coordinates>
$$
R=\sqrt{1+a}\,r,
\qquad
\Theta=\frac{\theta}{\sqrt{1+a}}
$$
turns this into the Euclidean metric $dR^2+R^2d\Theta^2$.
= Area element of a surface
{title2=$dA$}
{parent=First fundamental form}
{wiki=Surface_integral}
In a positively oriented surface chart,
$$
dA=\sqrt{EG-F^2}\,du\wedge dv.
$$
The coordinate-change Jacobian and the metric determinant transform inversely, so this defines a global area form.
= Vector area element
{title2=$d\mathbf S$}
{parent=Area element of a surface}
{wiki=Surface_integral}
For an oriented parametrized surface $\mathbf r(u,v)$,
$$
d\mathbf S=(\mathbf r_u\times\mathbf r_v)\,du\,dv
=\mathbf n\,dA.
$$
= Gauss map
{parent=Differential geometry}
{c}
{wiki}
For an oriented regular surface in $\mathbb R^3$, the Gauss map assigns its chosen unit normal
$$
N=\frac{X_u\times X_v}{|X_u\times X_v|}.
$$
= Second fundamental form
{title2=$II$}
{parent=Gauss map}
{wiki}
The second fundamental form is $II(v,w)=\langle-dN(v),w\rangle$. In a parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.
= Shape operator
{title2=$S_p$}
{parent=Second fundamental form}
{wiki=Shape_operator}
For an <orientation of a surface>[oriented surface] with <Gauss map> $N$, the shape operator at $p$ is the self-adjoint linear map
$$
S_p=-dN_p:T_pS\to T_pS.
$$
= Principal curvature
{title2=$k_1,k_2$}
{parent=Shape operator}
{wiki}
The principal curvatures are the two <eigenvalue>[eigenvalues] $k_1,k_2$ of the <shape operator>. Thus $K=k_1k_2$ and $H=(k_1+k_2)/2$.
= Umbilical point
{parent=Principal curvature}
{wiki=Umbilical_point}
An umbilical point of a surface is a point where the two <principal curvature>[principal curvatures] are equal, equivalently where the <shape operator> is a scalar multiple of the identity.
= Euclidean invariance of the shape operator
{parent=Shape operator}
For a <proper Euclidean motion of Euclidean three-space> $E(x)=Ax+b$, transport the <unit normal> by $\widetilde N(E(p))=AN(p)$. The corresponding shape operators are orthogonally conjugate:
$$
\widetilde S_{E(p)}=A S_p A^{-1}.
$$
Their <determinant> and <matrix trace>[trace], hence <Gaussian curvature> and <mean curvature>, are unchanged.
= Gaussian curvature
{title2=$K$}
{parent=Second fundamental form}
{c}
{wiki}
Gaussian curvature is the determinant of the shape operator:
$$
K=\frac{eg-f^2}{EG-F^2}.
$$
= Gaussian curvature of a cone away from its vertex
{parent=Gaussian curvature}
Every regular cone parametrized by $X(r,t)=r\,c(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, while
$$
X_{rr}=0,
\qquad
X_{rt}=c'(t)
$$
is tangent, so two coefficients of the <second fundamental form> vanish and $\det II=0$.
= Mean curvature
{title2=$H$}
{parent=Second fundamental form}
{wiki}
For a surface in $\mathbb R^3$, mean curvature is half the trace of the shape operator:
$$
H=\frac{eG-2fF+gE}{2(EG-F^2)}.
$$
= Minimal surface
{parent=Mean curvature}
{wiki}
A minimal surface has <mean curvature> $H=0$ at every point. Equivalently, its two <principal curvature>[principal curvatures] sum to zero.
= Orientation reversal of surface curvature
{parent=Mean curvature}
Replacing the chosen <unit normal> $N$ by $-N$ replaces the <shape operator> $S$ by $-S$. On a surface, <Gaussian curvature> $K=\det S$ is unchanged, while <mean curvature> $H=\operatorname{tr}(S)/2$ changes sign.
= Fundamental forms of a graph surface
{parent=Second fundamental form}
For $X(x,y)=(x,y,h(x,y))$ and $W=\sqrt{1+h_x^2+h_y^2}$,
$$
I=
\begin{pmatrix}1+h_x^2&h_xh_y\\h_xh_y&1+h_y^2\end{pmatrix},
\qquad
II=\frac1W
\begin{pmatrix}h_{xx}&h_{xy}\\h_{xy}&h_{yy}\end{pmatrix}.
$$
The upward unit normal is $(-h_x,-h_y,1)/W$.
= Gaussian curvature of a graph surface
{parent=Fundamental forms of a graph surface}
{c}
For the graph of $h$,
$$
K=\frac{h_{xx}h_{yy}-h_{xy}^2}{(1+h_x^2+h_y^2)^2}.
$$
= Mean-curvature comparison at tangential contact
{parent=Fundamental forms of a graph surface}
Suppose two graph surfaces $z=f(x,y)$ and $z=g(x,y)$ are tangent at the origin, are oriented upward, and $g\geq f$ nearby. Then $g-f$ has a local minimum, so its <Hessian matrix> is <positive semidefinite matrix>[positive semidefinite]. At the common horizontal tangent plane,
$$
H_g-H_f=\frac12\operatorname{tr}\operatorname{Hess}(g-f)\geq0.
$$
= Gaussian curvature has no tangential-contact comparison principle
{parent=Mean-curvature comparison at tangential contact}
The same contact does not order <Gaussian curvature>, because determinant is not monotone under addition of a positive-semidefinite matrix when the original Hessian is indefinite or negative definite. For example,
$$
f=-x^2-y^2,
\qquad
g=-\frac{x^2+y^2}{2}
$$
satisfy $g\geq f$ and are tangent at zero, but $K_g(0)=1<4=K_f(0)$.
= Tangency to a plane along a curve forces zero Gaussian curvature
{parent=Fundamental forms of a graph surface}
After a rigid motion, write the plane as $z=0$ and the surface locally as $z=h(x,y)$. Along the curve of tangency, $h=0$ and $\nabla h=0$. Differentiating $\nabla h(\gamma(s))=0$ shows that the Hessian annihilates the nonzero tangent $\gamma'(s)$, so its determinant and therefore the Gaussian curvature vanish.
= Minimal surface equation for a graph
{parent=Fundamental forms of a graph surface}
{wiki=Minimal_surface_equation}
The graph of $h$ is minimal exactly when
$$
(1+h_y^2)h_{xx}-2h_xh_yh_{xy}+(1+h_x^2)h_{yy}=0.
$$
Equivalently, it is the Euler--Lagrange equation
$$
\operatorname{div}\frac{\nabla h}{\sqrt{1+|\nabla h|^2}}=0
$$
for the area functional $\int\sqrt{1+|\nabla h|^2}$.
= Smooth curve
{parent=Differential geometry}
{wiki=Parametric_equation}
A smooth curve is a smooth map from an interval into a manifold or Euclidean space.
= Regular curve
{parent=Smooth curve}
{wiki=Regular_curve}
A smooth parametrized curve is regular when its velocity never vanishes.
= Arc-length parametrization
{parent=Regular curve}
{wiki=Arc_length\#Finding_arc_lengths_by_integration}
For a regular curve, $s(t)=\int_{t_0}^t|\alpha'(r)|dr$ has positive derivative and hence a smooth inverse. Reparametrization by $s$ gives unit speed.
= Curvature and torsion
{title2=$\kappa,\tau$}
{parent=Differential geometry}
{wiki=Curvature}
For unit speed, $\kappa=|T'|$ and $\tau=-B'\cdot N$. Generally $\kappa=|r'\times r''|/|r'|^3$ and $\tau=\det(r',r'',r''')/|r'\times r''|^2$.
= Curvature of a space curve
{title2=$\kappa$}
{parent=Curvature and torsion}
{wiki=Curvature#Space_curves}
For a unit-speed curve $\gamma$, the unit tangent is $T=\gamma'$ and the curvature is $\kappa=|T'|$. It measures the rate at which the tangent direction turns.
= Plane curve reconstructed from curvature
{parent=Curvature of a space curve}
Given a smooth function $\kappa(s)$, set
$$
\theta(s)=\int_0^s\kappa(u)\,du,
\qquad
\gamma(s)=\int_0^s(\cos\theta(u),\sin\theta(u))\,du.
$$
Then $\gamma$ is unit speed and has signed curvature $\kappa$. This gives the existence part of the fundamental theorem of plane curves.
= Total curvature
{parent=Curvature and torsion}
{wiki}
The total curvature of a regular curve is
$$
\int\kappa\,ds.
$$
It equals the length traced by the unit tangent on the unit sphere, so lifting a planar circle into a helix can reduce total curvature even when both curves make one revolution around the same axis.
= Torsion of a space curve
{parent=Curvature and torsion}
{wiki=Torsion_of_a_curve}
For a unit-speed curve with nonzero curvature, torsion is $\tau=-B'\cdot N$, measuring rotation of the osculating plane.
= Planar curves have zero torsion
{parent=Curvature and torsion}
All derivatives of a planar curve lie in one fixed two-dimensional vector plane, so $\det(r',r'',r''')=0$ and its torsion vanishes wherever defined.
= Steiner symmetrization
{parent=Differential geometry}
{c}
{wiki}
Steiner symmetrization replaces every chord perpendicular to a chosen axis by a centered chord of the same length.
= Cavalieri principle
{parent=Steiner symmetrization}
{c}
{wiki=Cavalieri%27s_principle}
Measurable planar sets with equal one-dimensional slice lengths in one direction have equal areas.
= Area preservation under Steiner symmetrization
{parent=Steiner symmetrization}
Every perpendicular chord keeps its length under Steiner symmetrization, so Fubini's theorem preserves the total area.
= Perimeter decrease under Steiner symmetrization
{parent=Steiner symmetrization}
For a convex planar domain between graphs $u_-$ and $u_+$, the Euclidean triangle inequality gives
$$
\sqrt{1+u_+'{}^2}+\sqrt{1+u_-'{}^2}
\geq\sqrt{4+(u_+'-u_-')^2},
$$
which is the pointwise perimeter comparison with its symmetrization.
= Equality case in the Steiner perimeter inequality
{parent=Perimeter decrease under Steiner symmetrization}
Equality holds exactly when $u_+'=-u_-'$, so the midpoints of all perpendicular chords lie on one line parallel to the symmetrizing axis.
= Symmetrization rigidity for a perimeter minimizer
{parent=Steiner symmetrization}
If Steiner symmetrization preserves area and cannot lower the perimeter of a minimizer, equality forces an axis of symmetry in the chosen direction. Applying every direction gives symmetry axes in all directions.
= Frenet-Serret formulas
{c}
{parent=Differential geometry}
{wiki=Frenet–Serret_formulas}
For a unit-speed space curve with nonzero curvature, $t=\dot\alpha$, $\kappa=|\dot t|$, $n=\dot t/\kappa$, and $b=t\times n$. With the torsion convention used here,
$$
\dot t=\kappa n,\qquad
\dot n=-\kappa t+\tau b,\qquad
\dot b=-\tau n.
$$
= Frenet frame
{title2=$(T,N,B)$}
{parent=Frenet-Serret formulas}
{wiki=Frenet–Serret_formulas#The_Frenet–Serret_frame}
For a sufficiently smooth unit-speed curve with $\kappa>0$, the Frenet frame is the positively oriented orthonormal frame
$$
T=\gamma',\qquad N=\frac{T'}{\kappa},\qquad B=T\times N.
$$
= Euclidean motion of Euclidean three-space
{title2=$E(3)$}
{parent=Frenet-Serret formulas}
{wiki=Euclidean_group}
A Euclidean motion has the form $E(x)=Ax+b$ with $A\in O(3)$. It preserves <Euclidean distance>, angles, and all extrinsic geometric quantities transformed with their orientations.
= Euclidean motion
{synonym}
= Proper Euclidean motion of Euclidean three-space
{title2=$\operatorname{SE}(3)$}
{parent=Euclidean motion of Euclidean three-space}
{wiki=Euclidean_group}
A proper Euclidean motion has the form $E(x)=Ax+b$ with $A\in SO(3)$. It preserves lengths, dot products, cross products, orientation, curvature, and torsion.
= Circular helix
{parent=Frenet-Serret formulas}
{wiki=Helix}
The unit-speed circular helix
$$
\gamma(s)=\left(a\cos\frac{s}{\sqrt{a^2+b^2}},
a\sin\frac{s}{\sqrt{a^2+b^2}},
\frac{bs}{\sqrt{a^2+b^2}}\right)
$$
has constant curvature $a/(a^2+b^2)$ and constant torsion $b/(a^2+b^2)$.
= Pointwise Euclidean invariant of a curve
{parent=Frenet-Serret formulas}
A scalar assignment $Q(\gamma,s)$ is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfies
$$
Q(\gamma_{s_0},s)=Q(\gamma,s-s_0)
$$
under translation of the arc-length parameter.
= Pointwise Euclidean invariants need not depend only on current curvature and torsion
{parent=Pointwise Euclidean invariant of a curve}
For sufficiently smooth curves, $Q(\gamma,s)=\kappa'(s)$ obeys Euclidean and parameter-translation covariance but is not determined by the pair $(\kappa(s),\tau(s))$. Under the bare covariance definition, nonlocal examples such as $Q(\gamma,s)=\kappa(s+1)$ work as well.
= Curvature decomposition for a spherical curve
{parent=Frenet-Serret formulas}
For a unit-speed curve on the unit sphere,
$$
\alpha=-\kappa^{-1}n-\tau^{-1}\kappa^{-2}\dot\kappa\,b,
\qquad
\kappa_g=-\kappa^{-1}\tau^{-1}\dot\kappa
$$
under the compatible signed-torsion and geodesic-curvature conventions.
= Geodesic curvature
{parent=Differential geometry}
{wiki}
For a unit-speed curve $\alpha$ on an oriented surface with unit normal $N$, let $T=\dot\alpha$ and let $D_s$ denote tangential covariant differentiation. Its signed geodesic curvature is
$$
\kappa_g=\langle D_sT,N\times T\rangle,
\qquad
D_sT=\kappa_g(N\times T).
$$
It vanishes exactly when the curve is a geodesic.
= Transitive curve symmetry makes geodesic-curvature magnitude constant
{parent=Geodesic curvature}
If ambient isometries preserving a connected curve act transitively on it, then the magnitude of its geodesic curvature is constant. Continuity then makes the signed geodesic curvature either identically zero or of one fixed sign.
= Isometric halves give zero total boundary geodesic curvature
{parent=Geodesic curvature}
If a closed oriented surface is cut along a curve into two isometric surfaces with boundary, their Euler characteristics and total Gaussian curvatures agree. Applying Gauss-Bonnet to both halves, whose induced boundary orientations are opposite, gives
$$
\int_\gamma k_g\,ds=0.
$$
= Homogeneous nongeodesic latitude
{parent=Geodesic curvature}
A non-equatorial latitude on the round sphere is preserved transitively by rotations about its axis but has nonzero geodesic curvature. Its two complementary spherical caps are not isometric.
= Antipodally symmetric nongeodesic spherical curve
{parent=Geodesic curvature}
For sufficiently small nonzero $\varepsilon$,
$$
\gamma(\theta)=
\frac{(\cos\theta,\sin\theta,\varepsilon\sin3\theta)}
{\sqrt{1+\varepsilon^2\sin^23\theta}}
$$
is an embedded antipodally invariant curve on the round sphere. It is not a great circle and hence not a geodesic, while the antipodal isometry exchanges its two complementary discs.
= Gauss-Bonnet theorem
{c}
{parent=Differential geometry}
{wiki=Gauss–Bonnet_theorem}
$$
\int_DK\,dA+\int_{\partial D}k_g\,ds+\sum\text{ exterior angles}=2\pi\chi(D).
$$
= Area of a hyperbolic geodesic polygon
{parent=Gauss-Bonnet theorem}
In the curvature-minus-one <Poincare half-plane model>, a geodesic polygon with $n$ sides and interior angles $\alpha_1,\ldots,\alpha_n$ has area
$$
(n-2)\pi-\sum_{j=1}^n\alpha_j.
$$
This is the polygonal <Gauss-Bonnet theorem>.
= Spherical isoperimetric inequality
{parent=Gauss-Bonnet theorem}
A simple closed curve of length $L$ on the unit sphere enclosing the smaller area $A$ satisfies
$$
L^2\geq A(4\pi-A).
$$
Equality holds for a circle. A variational proof uses Gauss--Bonnet to turn area maximization at fixed length into geodesic-curvature minimization, proves that an extremizer has constant geodesic curvature and is planar, and evaluates its spherical cap.
= Gaussian curvature of a torus
{parent=Gauss-Bonnet theorem}
{c}
{wiki}
For a standard ring torus, Gaussian curvature is positive outside, negative inside, and zero along the top and bottom circles.
= Total Gaussian curvature
{parent=Gauss-Bonnet theorem}
{wiki}
The total Gaussian curvature of a closed surface is two pi times its Euler characteristic.
= Intersection of closed geodesics on a positively curved sphere
{parent=Gauss-Bonnet theorem}
Two disjoint simple closed curves on a sphere bound an annulus $A$. If both are geodesics, its boundary geodesic-curvature terms vanish, whereas $\chi(A)=0$. Gauss-Bonnet would give
$$
\int_AK\,dA=0,
$$
which is impossible when $K>0$ everywhere.
= Preimage theorem
{parent=Differential geometry}
{wiki}
The inverse image of a regular value of a smooth map is a submanifold of codimension equal to the target dimension.
= Theorema Egregium
{c}
{parent=Differential geometry}
{wiki}
Gaussian curvature depends only on the first fundamental form and is therefore preserved by local isometries.
= Riemannian geometry
{parent=Differential geometry}
{wiki}
Riemannian geometry studies smooth manifolds equipped with smoothly varying inner products on tangent spaces.
= Riemannian surface
{parent=Riemannian geometry}
{wiki}
A Riemannian surface is a two-dimensional smooth manifold equipped with a smoothly varying inner product on each tangent plane.
= Arc length
{title2=$L(\gamma)$}
{parent=Riemannian geometry}
{wiki}
The arc length of a regular curve $\gamma:[a,b]\to M$ is
$$
L(\gamma)=\int_a^b|\dot\gamma(t)|\,dt.
$$
= Isometry
{parent=Riemannian geometry}
{wiki}
An isometry preserves the Riemannian metric and therefore distances, angles, geodesics, and intrinsic curvature.
= Geodesic
{parent=Riemannian geometry}
{wiki}
A geodesic is a curve whose tangent vector is parallel along the curve. Equivalently, it has zero covariant acceleration; on an embedded surface, a unit-speed geodesic has acceleration normal to the surface.
= Geodesic triangle
{parent=Geodesic}
{wiki=Geodesic_polygon}
A geodesic triangle is bounded by three geodesic segments. On a surface its angle excess is the integral of the <Gaussian curvature> over the triangle.
= Surface covariant derivative
{parent=Geodesic}
{wiki=Covariant_derivative}
For a tangent vector field $V$ along a curve on an embedded surface, the surface covariant derivative is the tangential projection of its ordinary derivative:
$$
D_sV=\left(\frac{dV}{ds}\right)^\top.
$$
It is the covariant derivative induced by the surface metric.
= Exponential map
{title2=$\exp_p$}
{parent=Geodesic}
{wiki=Exponential_map_(Riemannian_geometry)}
For $p$ in a Riemannian manifold and $v$ sufficiently close to zero in $T_pM$, let $\gamma_v$ be the geodesic with $\gamma_v(0)=p$ and $\dot\gamma_v(0)=v$. The exponential map is
$$
\exp_p(v)=\gamma_v(1).
$$
Equivalently, $\exp_p(tv)=\gamma_v(t)$ wherever both sides are defined.
= Domain of the exponential map
{parent=Exponential map}
The domain of $\exp_p$ is the star-shaped open set
$$
\mathcal D_p=\{v\in T_pM:\gamma_v\text{ exists on }[0,1]\}.
$$
Smooth dependence for ordinary differential equations makes $(p,v)\mapsto\exp_p(v)$ smooth on its open domain in $TM$.
= Exponential map of the punctured plane
{parent=Domain of the exponential map}
For $S=(\mathbb R^2\setminus\{0\})\times\{0\}\subset\mathbb R^3$ and $p=(1,0,0)$, geodesics are straight lines until they hit the missing origin. Thus $(-2,0,0)\notin\mathcal D_p$, and $(-1,0,0)$ is not in the image of $\exp_p$ because its unique straight segment from $p$ passes through the origin.
= Geodesic polar coordinates
{parent=Exponential map}
{wiki}
Choose an orthonormal basis of $T_pM$ and put $e(\theta)=(\cos\theta,\sin\theta)$. Away from the centre and cut locus,
$$
\phi(r,\theta)=\exp_p\bigl(r e(\theta)\bigr)
$$
defines geodesic polar coordinates centred at $p$.
= Gauss lemma
{parent=Geodesic polar coordinates}
{c}
{wiki=Gauss%27s_lemma_(Riemannian_geometry)}
The radial direction is orthogonal to the angular directions under the exponential map. On a surface this makes the first fundamental form in geodesic polar coordinates
$$
I=dr^2+G(r,\theta)\,d\theta^2,
\qquad
G=|\phi_\theta|^2,
$$
with $\sqrt{G(r,\theta)}/r\to1$ as $r\to0$.
= Complete geodesic
{parent=Geodesic}
{wiki=Geodesic\#Examples}
A geodesic is complete when its maximal affine parameter interval is all of $\mathbb R$.
= Hopf-Rinow theorem
{c}
{parent=Complete geodesic}
{wiki=Hopf–Rinow_theorem}
Every connected compact Riemannian manifold is geodesically complete. In particular, every exponential map is defined on its full tangent space.
= Tangency invariance of an ambient-surface geodesic
{parent=Geodesic}
If two embedded surfaces are tangent along a curve, their tangent planes agree there. The curve's ambient acceleration is normal to one surface exactly when it is normal to the other, so it is a geodesic of one exactly when it is a geodesic of the other.
= Cylindrical-helix tangency construction
{parent=Tangency invariance of an ambient-surface geodesic}
A circular helix on a circular cylinder has radial ambient acceleration and is therefore a geodesic. Any surface tangent to the cylinder along that helix has the same curve as a geodesic.
= Energy of a curve
{parent=Riemannian geometry}
The energy of a curve is one half the integral of its squared speed.
= Christoffel symbol
{title2=$\Gamma^k_{ij}$}
{parent=Riemannian geometry}
{c}
{wiki}
Christoffel symbols are the coordinate coefficients of the Levi-Civita connection.
= Geodesic equation
{parent=Riemannian geometry}
{wiki}
A geodesic has zero covariant acceleration and in coordinates satisfies $\ddot u^k+\Gamma^k_{ij}\dot u^i\dot u^j=0$.
= Ambient acceleration criterion for a surface geodesic
{parent=Geodesic equation}
For a smoothly parametrized curve on an embedded surface in Euclidean space, the geodesic equations hold exactly when the ambient acceleration is normal to the surface.
= Constant speed of an affinely parametrized geodesic
{parent=Ambient acceleration criterion for a surface geodesic}
The velocity of a surface curve is tangent. If its acceleration is normal, then
$$
\frac d{dt}|\dot\gamma|^2=2\dot\gamma\mathbin{\cdot}\ddot\gamma=0,
$$
so every affinely parametrized geodesic has constant speed.
= Normal section of a surface
{parent=Geodesic equation}
A constant-speed intersection with a plane containing every surface normal along the curve is a geodesic.
= Affine parameter
{parent=Geodesic equation}
{wiki=Geodesic#Affine_geodesic}
An affine parameter on a geodesic is one for which the coordinate equation has the form $\ddot x^k+\Gamma^k_{ij}\dot x^i\dot x^j=0$. Its affine changes $\lambda\mapsto A\lambda+B$, with $A\ne0$, preserve this form.
= Embedded surface parametrization
{parent=Differential geometry}
{wiki}
An embedded-surface chart is a homeomorphic smooth parametrisation with derivative of rank two.
= Surface of revolution
{parent=Differential geometry}
{wiki}
A surface of revolution is invariant under rotations around a fixed axis and is locally generated by rotating a profile curve.
= Catenoid
{parent=Surface of revolution}
{wiki}
A catenoid is the surface of revolution obtained by rotating a <catenary> about its directrix. One conformal parametrization is
$$
(u,v)\mapsto(\cosh u\cos v,\cosh u\sin v,u).
$$
= Constant strip-area density of a surface of revolution
{parent=Surface of revolution}
For
$$
X(\theta,z)=(\phi(z)\cos\theta,\phi(z)\sin\theta,z),
$$
the area density after integrating over $\theta$ is $2\pi\phi\sqrt{1+\phi'^2}$. If every height interval has area $2\pi r$ times its length, continuity forces
$$
\phi^2(1+\phi'^2)=r^2.
$$
Where $0<\phi<r$, the sign of $\phi'$ is constant and $\sqrt{r^2-\phi^2}$ has derivative $\pm1$. Hence the profile lies on a circle of radius $r$.
= Curvature-matching diffeomorphism between two surfaces of revolution
{parent=Surface of revolution}
For the surfaces generated by $(e^u,0,u)$ and $(\cosh s,0,s)$,
$$
K_R(u)=-(1+e^{2u})^{-2},
\qquad
K_S(s)=-\cosh^{-4}s.
$$
The diffeomorphism $u=\log\sinh s$ matches these curvatures, although comparison of the first fundamental forms shows that no local isometry exists.
= First fundamental form of an arc-length surface of revolution
{parent=Surface of revolution}
For $X(u,v)=(Y(u)\cos v,Y(u)\sin v,Z(u))$ with $Y'^2+Z'^2=1$, the induced metric is
$$
du^2+Y(u)^2\,dv^2.
$$
= Curvatures of an arc-length surface of revolution
{parent=First fundamental form of an arc-length surface of revolution}
With unit normal $(-Z'\cos v,-Z'\sin v,Y')$, the principal-form coefficients give
$$
K=-\frac{Y''}{Y},
\qquad
H=\frac12\left(Y'Z''-Z'Y''+\frac{Z'}Y\right).
$$
= Constant Gaussian curvature surfaces of revolution
{parent=Curvatures of an arc-length surface of revolution}
For an arc-length profile with radius $Y(u)>0$, the equation $K=c$ is
$$
Y''+cY=0,
\qquad
Z'=\sqrt{1-Y'^2}.
$$
Thus $Y=A\cos u$ gives $K=1$ locally, while $Y=A\cosh u$ gives $K=-1$ wherever $|Y'|<1$. At $u=0$ their mean curvatures are respectively
$$
\frac12\left(A+\frac1A\right)
\quad\hbox{and}\quad
\frac12\left(-A+\frac1A\right),
$$
so different choices of $A$ give locally noncongruent surfaces with the same constant Gaussian curvature.
= Gaussian curvature of a surface of revolution
{parent=Curvatures of an arc-length surface of revolution}
{c}
For the parametrization
$$
X(x,\theta)=(x,f(x)\cos\theta,f(x)\sin\theta),
$$
the Gaussian curvature and area element are
$$
K=-\frac{f''}{f(1+f'^2)^2},
\qquad
dA=f\sqrt{1+f'^2}\,dx\,d\theta.
$$
= Total Gaussian curvature of a surface-of-revolution strip
{parent=Gaussian curvature of a surface of revolution}
For $a\leq x\leq b$, the curvature formula is an exact derivative and gives
$$
\int K\,dA
=-2\pi\left[
\frac{f'}{\sqrt{1+f'^2}}
\right]_{a}^{b}.
$$
= Circular cylinder
{parent=Surface of revolution}
{wiki=Cylinder}
A circular cylinder of radius $R$ has <principal curvature>[principal curvatures] $0$ and $\pm1/R$, according to orientation. Hence $K=0$ and $|H|=1/(2R)$.
= Clairaut first integral for a surface of revolution
{parent=Surface of revolution}
{c}
For a geodesic in the metric $du^2+Y(u)^2dv^2$, the cyclic coordinate $v$ gives
$$
Y(u)^2\dot v=\text{constant}.
$$
Equivalently, the geodesic meets parallels according to Clairaut's relation.
= Clairaut's relation
{c}
{synonym}
= Helicoid
{parent=Differential geometry}
{wiki}
A helicoid is a ruled minimal surface swept out by a line that rotates while translating along its perpendicular axis.
= Ruled surface
{parent=Differential geometry}
{wiki}
A ruled surface is swept out by a smoothly varying one-parameter family of affine lines.
= One-sheet hyperboloid
{parent=Ruled surface}
{wiki}
A one-sheet hyperboloid is a connected doubly ruled quadric, diffeomorphic to a cylinder.
= Hyperboloid
{synonym}
= Plane-section geodesics of the unit one-sheet hyperboloid
{parent=One-sheet hyperboloid}
On $x^2+y^2=z^2+1$, an axial plane such as $y=0$ cuts out two disjoint meridian geodesics. The tangent plane $x=1$ cuts out the two straight ruling lines $y=\pm z$, which are geodesics meeting orthogonally at $(1,0,0)$.
= Geodesic trapped in one half of the unit one-sheet hyperboloid
{parent=One-sheet hyperboloid}
For a unit-speed geodesic on $x^2+y^2=z^2+1$, Clairaut's constant $c$ gives
$$
\dot z^2=\frac{z^2+1-c^2}{1+2z^2}.
$$
If $c>1$, the geodesic has a turning point at $z=\sqrt{c^2-1}$ and remains entirely in one of the regions $z>0$ or $z<0$.
= Isometry-invariant waist geodesic of the unit one-sheet hyperboloid
{parent=One-sheet hyperboloid}
The waist circle $z=0$ is a geodesic. Its Gaussian curvature is $-1$, whereas
$$
K(z)=-\frac1{(1+2z^2)^2}>-1
$$
off the waist. Since isometries preserve Gaussian curvature, every isometry preserves this circle setwise.
= Flat cone
{parent=Differential geometry}
{wiki}
A cone is intrinsically flat away from its vertex but has nontrivial angular holonomy.
= Developing map
{parent=Flat cone}
{wiki}
A developing map locally unfolds a flat surface into the Euclidean plane.
= Isometry of a cone
{parent=Flat cone}
{wiki}
Cone isometries preserve its angular identification; those fixing a point are the identity or an axial-plane reflection.
= Manifold chart
{parent=Differential geometry}
{wiki=Manifold\#Charts}
A chart is a homeomorphism from an open subset of a manifold to an open subset of Euclidean space.
= No compact manifold has a single Euclidean chart
{parent=Manifold chart}
A nonempty compact manifold cannot be homeomorphic to an open subset of Euclidean space, since no nonempty Euclidean open set is compact.
= Regular value
{parent=Differential geometry}
{wiki=Regular_value}
A value is regular when the derivative is surjective at every point of its preimage.
= Regular level set theorem
{parent=Regular value}
{wiki=Preimage_theorem}
The preimage of a regular value of a smooth map between manifolds is a smooth submanifold whose codimension equals the dimension of the target.
= Sard theorem
{c}
{parent=Regular value}
{wiki=Sard%27s_theorem}
For a smooth map between smooth manifolds, the set of critical values has measure zero in the target. In particular, regular values are dense.
= Degree modulo two
{title2=$\deg_2$}
{parent=Differential geometry}
{wiki=Degree_of_a_continuous_mapping}
For a smooth map $f:X\to Y$ between compact connected $n$-manifolds and a regular value $y$, define
$$
\deg_2(f)=|f^{-1}(y)|\pmod2.
$$
The regular-value theorem makes the preimage discrete and compact, hence finite. A transverse path between regular values produces a compact one-manifold whose boundary is the two fibres, proving independence of $y$. The degree is invariant under smooth homotopy.
= Exponential displacement map on a compact surface
{parent=Degree modulo two}
If $S$ is compact and $V$ is a smooth vector field, then
$$
\phi(p)=\exp_p(V(p))
$$
is smooth and well-defined by the <Hopf-Rinow theorem>. The maps $\phi_t(p)=\exp_p(tV(p))$ form a smooth homotopy from the identity to $\phi$, so $\deg_2(\phi)=1$.
= Symplectic group
{title2=$Sp(2n,\mathbb R)$}
{parent=Differential geometry}
{wiki}
The real symplectic group consists of matrices $A$ satisfying $A^TJA=J$ and has dimension $2n^2+n$ in size $2n$.
= Tangent space of the symplectic group
{parent=Symplectic group}
At the identity, the symplectic Lie algebra consists of $X$ satisfying $X^TJ+JX=0$; tangent spaces elsewhere are its left translates.
= Grassmannian as projection matrices
{parent=Differential geometry}
{wiki=Grassmannian}
The real Grassmannian of $k$-planes is represented by symmetric idempotent matrices of trace $k$.
= Rank-one orthogonal projection
{parent=Grassmannian as projection matrices}
Every rank-one orthogonal projection has the form $xx^T$ for a unit vector, uniquely up to replacing $x$ by $-x$.
= Real projective plane
{parent=Grassmannian as projection matrices}
{wiki}
The real projective plane is the quotient of $S^2$ by $x\sim-x$ and parametrizes lines through the origin in $\mathbb R^3$.
= Classification theorem for surfaces
{parent=Geometry and topology}
{wiki}
= Conformal map
{parent=Geometry and topology}
{wiki}
= Conformal equivalence
{parent=Conformal map}
{wiki}
Two plane domains are conformally equivalent when there is a bijective holomorphic map between them whose inverse is holomorphic.
= Conformally equivalent
{synonym}
= Punctured plane is not conformally equivalent to the punctured unit disc
{parent=Conformal equivalence}
A conformal map from $\mathbb C^*$ into the punctured unit disc would be bounded near its isolated singularity at zero, hence extend to a bounded entire function. The <Liouville theorem> would make it constant.
= Complex plane is not conformally equivalent to the upper half-plane
{parent=Conformal equivalence}
Composing a hypothetical conformal equivalence $\mathbb C\to\mathbb H$ with a <Cayley transform> $\mathbb H\to\mathbb D$ would give a nonconstant bounded <entire function>, contradicting the <Liouville theorem>.
= Conformal equivalence between the upper half-disc and upper half-plane
{parent=Conformal equivalence}
The map
$$
z\longmapsto\left(\frac{1+z}{1-z}\right)^2
$$
maps $\{z:|z|<1,\ \operatorname{Im}z>0\}$ conformally onto the upper half-plane. The <Möbius transformation> maps the half-disc onto the first quadrant, and squaring maps that quadrant onto the upper half-plane.
= Holomorphic inverse function theorem
{parent=Conformal map}
{wiki=Inverse_function_theorem\#Holomorphic_functions}
If a holomorphic function $f$ satisfies $f'(z_0)\ne0$, then it has a holomorphic local inverse near $z_0$ and is conformal there. Its possible failures of local conformality are therefore its critical points.
= Hyperbolic-cosine half-strip map
{parent=Conformal map}
The map $z\mapsto\cosh(\pi z)$ is a conformal bijection from
$$
\{z:\Re z>0,\ 0<\Im z<1\}
$$
onto the upper half-plane. Its three boundary intervals map respectively to $(1,\infty)$, $(-1,1)$ and $(-\infty,-1)$.
= Connected space
{parent=Geometry and topology}
{wiki}
A connected space cannot be expressed as the union of two disjoint nonempty open subsets.
= Connected component
{parent=Connected space}
{wiki}
A connected component is a maximal connected subset of a topological space.
= Jordan-Brouwer separation theorem
{c}
{parent=Connected component}
{wiki}
An embedded copy of $S^{n-1}$ in $\mathbb R^n$ separates space into an inside and an outside. In particular, a connected closed surface embedded in $\mathbb R^3$ is two-sided and orientable.
= Connected
{synonym}
= Continuous image of a connected space
{parent=Connected space}
{wiki}
A continuous image of a connected space is connected.
= Integer-valued function criterion for connectedness
{parent=Connected space}
A topological space is connected exactly when every continuous map from it to the discrete space $\mathbb Z$ is constant. A disconnection produces a nonconstant indicator map, while the continuous image of a connected space in $\mathbb Z$ must be a singleton.
= Pairwise-intersecting connected cover
{parent=Connected space}
If connected subsets cover a space and every two of them intersect, then their union is connected. Equivalently, any continuous integer-valued function is constant on each member, and the intersections force all those constants to agree.
= Product of connected spaces
{parent=Connected space}
{wiki=Connected_space\#Products_and_quotients}
The product of any family of connected spaces is connected. For two factors, fix $y_0$ and use the connected sets
$$
(X\times\{y_0\})\cup(\{x\}\times Y),
\qquad x\in X.
$$
They cover $X\times Y$ and all contain the connected horizontal slice $X\times\{y_0\}$.
= Closure of a connected set
{parent=Connected space}
{wiki}
Every set lying between a connected set and its closure is connected.
= Distance from a point to a plane
{parent=Geometry and topology}
{wiki}
= Elliptic coordinates
{parent=Geometry and topology}
{wiki}
= Joukowski map
{parent=Geometry and topology}
{c}
{wiki}
= Path-connected space
{parent=Geometry and topology}
{wiki}
A space is path connected when every two points can be joined by a continuous path within the space.
= Continuous path
{parent=Path-connected space}
{wiki=Path_(topology)}
A continuous path in $X$ is a continuous map $\gamma:[0,1]\to X$.
= Path
{synonym}
= Path component
{parent=Path-connected space}
{wiki=Connected_component#Path_connectedness}
A path component is a maximal path-connected subset of a topological space.
= Locally path-connected space
{parent=Path-connected space}
{wiki=Locally_connected_space}
A space is locally path-connected when every point has a neighbourhood basis of path-connected open sets. Its path components are open.
= Cut point
{parent=Path-connected space}
{wiki}
A cut point of a connected space is a point whose removal disconnects the space. Every interior point of a closed interval is a cut point, whereas a two-dimensional closed disc has no cut point.
= Plane
{parent=Geometry and topology}
{wiki}
= Plane Poiseuille flow
{parent=Geometry and topology}
{wiki}
= Product topology
{parent=Geometry and topology}
{wiki}
= Sequentially compact space
{parent=Geometry and topology}
{wiki}
A space is sequentially compact when every sequence has a convergent subsequence. A metric space is compact exactly when it is sequentially compact.
= Sphere-plane intersection
{parent=Geometry and topology}
{wiki}
= Spherically symmetric solution of Poisson equation
{parent=Geometry and topology}
{wiki}
= Topological degree
{title2=$\deg f$}
{parent=Geometry and topology}
{wiki}
Topological degree is an integer-valued signed count of preimages that is stable under suitable perturbations.
= Trigonometric function
{parent=Geometry and topology}
{wiki}
= Sine
{title2=$\sin$}
{parent=Trigonometric function}
{wiki}
= Cosine
{title2=$\cos$}
{parent=Trigonometric function}
{wiki}
= Tangent
{title2=$\tan$}
{parent=Trigonometric function}
{wiki=Tangent_(trigonometry)}
= Half-angle identities
{parent=Trigonometric function}
{wiki=Tangent_half-angle_substitution}
The half-angle identities include
$$
\tan\frac\theta2=\frac{1-\cos\theta}{\sin\theta}
=\frac{\sin\theta}{1+\cos\theta}.
$$
= Inverse trigonometric function
{parent=Trigonometric function}
{wiki=Inverse_trigonometric_functions}
= Inverse sine
{title2=$\arcsin$}
{parent=Inverse trigonometric function}
{wiki=Inverse_trigonometric_functions}
= Small-angle approximation
{parent=Trigonometric function}
{wiki=Small-angle_approximation}
As $x\to0$, the <limit> $\sin x/x\to1$ gives $\sin x\sim x$, while the <Taylor series> gives $\arcsin x\sim x$ and $\cos x\sim1-x^2/2$.
= Fractal geometry
{parent=Geometry and topology}
{wiki=Fractal}
Fractal geometry studies recursively structured sets and spaces, often with noninteger scaling dimension.
= Self-similarity
{parent=Fractal geometry}
{wiki=Self-similarity}
A self-similar object is assembled from smaller copies of itself, possibly after scaling or other transformations.
= Sierpinski triangle
{c}
{parent=Self-similarity}
{wiki=Sierpi%C5%84ski_triangle}
The Sierpinski triangle is formed recursively by retaining the three corner subtriangles after subdividing an equilateral triangle into four congruent pieces.
= Euclidean geometry
{parent=Geometry and topology}
{wiki}
Euclidean geometry studies distances, angles, and rigid figures in flat space.
= Unit disk
{parent=Euclidean geometry}
{wiki}
The open unit disk in the <Euclidean geometry>[Euclidean plane] is
$$
\{(x,y)\in\mathbb R^2:x^2+y^2<1\}.
$$
= Planar isoperimetric inequality
{parent=Euclidean geometry}
{wiki=Isoperimetric_inequality}
Every bounded planar domain with a sufficiently regular boundary of length $L$ and area $A$ satisfies
$$
4\pi A\leq L^2.
$$
Equality holds exactly for circular domains.
= Cyclic quadrilateral
{parent=Euclidean geometry}
{wiki=Cyclic_quadrilateral}
A cyclic quadrilateral has all four vertices on one circle. Among quadrilaterals with fixed cyclically ordered side lengths, a cyclic quadrilateral has maximal area.
= Bretschneider formula
{parent=Cyclic quadrilateral}
{c}
{wiki=Bretschneider%27s_formula}
For a quadrilateral with side lengths $a,b,c,d$, semiperimeter $s$, area $K$, and opposite angles $A,C$,
$$
K^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\frac{A+C}{2}.
$$
The second term vanishes exactly when $A+C=\pi$, which is the cyclicity condition for a nondegenerate convex quadrilateral.
= Ellipse
{parent=Euclidean geometry}
{wiki}
An ellipse is the locus of points whose distances from two fixed foci have constant sum. If the focal distance is $d$ and the sum is $c>d$, its semimajor and semiminor axes are $c/2$ and $\sqrt{c^2-d^2}/2$.
= Nondegenerate ellipse
{parent=Ellipse}
A nondegenerate ellipse has two positive semiaxis lengths. It is a smooth closed curve rather than a point or line segment.
= Focal line
{parent=Ellipse}
The focal line of an <ellipse> is the straight line through its two foci.
= Perpendicular bisector
{parent=Euclidean geometry}
{wiki}
The perpendicular bisector of a line segment is the line through its midpoint at a right angle to the segment.
= Apollonius circle
{c}
{parent=Euclidean geometry}
{wiki=Circles_of_Apollonius}
An Apollonius circle is the locus of points whose distances from two fixed points have a prescribed positive ratio different from one. Ratio one gives the perpendicular bisector.
= Right angle
{parent=Euclidean geometry}
{wiki}
A right angle is an angle of $\pi/2$ radians; two nonzero vectors meet at a right angle exactly when their inner product is zero.
= Rigid motion
{parent=Euclidean geometry}
{wiki=Rigid_transformation}
A rigid motion preserves Euclidean distances and therefore preserves angles, lengths, and curvatures.
= Thales theorem
{parent=Euclidean geometry}
{c}
{wiki=Thales%27s_theorem}
An angle subtended by a diameter of a circle is right; conversely, the hypotenuse of a right triangle inscribed in a circle is a diameter.
= Regular polygon
{parent=Euclidean geometry}
{wiki}
A regular polygon has equal side lengths and equal interior angles. Its vertices on a circle are scaled and rotated roots of unity.
= Area of a regular octagon
{parent=Regular polygon}
{wiki}
A regular octagon of side length s has area 2(1+sqrt(2))s^2.
= Spherical cap
{parent=Euclidean geometry}
{wiki}
A spherical cap is the part of a sphere cut off by a plane.
= Lattice polygon
{parent=Euclidean geometry}
{wiki}
A lattice polygon has all vertices in the integer lattice; triangulation shows that its area is a half-integer.
= Spherical symmetry
{parent=Geometry and topology}
{wiki}
Spherical symmetry means invariance under every orthogonal transformation fixing the centre.
= Isotropic tensor integral
{parent=Spherical symmetry}
{wiki}
A rotationally invariant rank-two tensor is a scalar multiple of $\delta_{ij}$; taking its trace determines the scalar.
= Differential form
{parent=Geometry and topology}
{wiki}
A differential form is an alternating covariant tensor field designed for coordinate-independent integration.
= Exact differential
{parent=Differential form}
{wiki}
An exact differential is the differential $df$ of a scalar potential and integrates to zero around every closed curve.
= Topology
{parent=Geometry and topology}
{wiki}
Topology studies spaces and properties preserved by continuous deformation.
= Closed set
{parent=Topology}
{wiki}
A closed set contains all its <limit points>; equivalently, its complement is <open set>[open].
= Closed sets
{synonym}
= Limit point
{parent=Closed set}
{wiki}
A limit point of a set is a point whose every neighbourhood contains a different point of the set.
= Limit points
{synonym}
= Closure
{title2=$\overline A$}
{parent=Topology}
{wiki=Closure_(topology)}
The closure of a subset is the smallest closed set containing it, equivalently the set of points whose every neighbourhood meets it.
= Topological space
{parent=Topology}
{wiki}
A topological space is a set together with a family of open subsets containing the empty set and the whole set and closed under arbitrary unions and finite intersections.
= Topology axiom
{parent=Topological space}
A family $\mathcal T$ of <subset>[subsets] of a <set> $X$ is a topology when $\varnothing,X\in\mathcal T$, every <union> of members of $\mathcal T$ belongs to $\mathcal T$, and every finite <intersection> of members of $\mathcal T$ belongs to $\mathcal T$.
= Sierpinski space
{c}
{parent=Topological space}
{wiki}
The Sierpinski space is the two-point space with exactly one open singleton.
= Open cover
{parent=Topological space}
{wiki}
An open cover of a subset $A$ is a family of open sets whose union contains $A$.
= Quotient topology
{parent=Topology}
{wiki}
For an equivalence relation $R$ on a topological space $X$, the quotient topology on the set $X/R$ declares $U\subseteq X/R$ open exactly when its inverse image under the projection $q:X\to X/R$ is open in $X$.
= Quotient map
{parent=Quotient topology}
{wiki}
A quotient map is a surjective continuous map $q:X\to Y$ for which $U\subseteq Y$ is open exactly when $q^{-1}(U)$ is open.
= Universal property of the quotient topology
{parent=Quotient topology}
If $q:X\to X/R$ is the quotient map and a continuous map $f:X\to Y$ is constant on each equivalence class, there is a unique continuous map $F:X/R\to Y$ satisfying $Fq=f$. It is given by $F([x])=f(x)$; continuity follows because
$$
q^{-1}(F^{-1}(U))=f^{-1}(U)
$$
for every open $U\subseteq Y$.
= Non-Hausdorff quotient of the real line by rational translation
{parent=Quotient topology}
Identify $x,y\in\mathbb R$ when $x-y\in\mathbb Q$. Every nonempty saturated open subset of $\mathbb R$ meets every other one because rational translates of any open interval meet any prescribed nonempty open interval. The quotient $\mathbb R/\mathbb Q$ has multiple points but no two nonempty disjoint open subsets, so it is not Hausdorff.
= Square quotient model of the two-sphere
{parent=Quotient topology}
Identify the vertical sides of $[0,1]^2$, then collapse each horizontal side separately to a point. The resulting quotient is homeomorphic to $S^2$ through
$$
(x,y)\longmapsto
\bigl(\sin(\pi y)\cos(2\pi x),
\sin(\pi y)\sin(2\pi x),\cos(\pi y)\bigr).
$$
Equivalently, the quotient is the suspension of the circle.
= Open set
{parent=Topology}
{wiki}
An open set is a member of the topology on a topological space.
= Open
{synonym}
= Open interval
{parent=Open set}
{wiki=Interval_(mathematics)}
An open interval in the real line is a set $(a,b)=\{x:a<x<b\}$, allowing either endpoint to be infinite.
= Nondegenerate interval
{parent=Open interval}
A nondegenerate interval contains more than one point; for an interval with finite endpoints, this means that its endpoints are distinct.
= Neighbourhood
{parent=Open set}
{wiki=Neighbourhood_(mathematics)}
A neighbourhood of a point is a set containing an open set that contains that point.
= Open ball
{parent=Open set}
{wiki=Ball_(mathematics)}
In a metric space, the open ball of radius $r$ about $x$ is $\{y:d(x,y)<r\}$.
= Open disc
{parent=Open set}
{wiki=Disk_(mathematics)}
The open disc with centre $z$ and radius $r>0$ is $\{w:|w-z|<r\}$.
= Unit disc
{title2=$\mathbb D$}
{parent=Open disc}
{wiki=Unit_disk}
The unit disc in the <complex plane> is $\mathbb D=\{z\in\mathbb C:|z|<1\}$.
= Disc
{title2=$D^n$}
{parent=Open disc}
{wiki=Disk_(mathematics)}
A disc consists of the points whose distance from a centre is below a fixed radius, with the boundary included or excluded according to context.
= Annulus
{parent=Open set}
{wiki=Annulus_(mathematics)}
An annulus is the region between two concentric circles.
= Fixed-point-free rotation of an annulus
{parent=Annulus}
Rotation through any nonzero angle defines a continuous self-map of a circular annulus with no fixed point. Hence an annulus does not have the <fixed-point property>.
= Level set
{parent=Topology}
{wiki}
A level set of a function $f$ is a set of the form $\{x:f(x)=c\}$.
= Dense subset
{parent=Topology}
{wiki=Dense_set}
A subset is dense when every nonempty <open set> meets it, equivalently when its closure is the whole ambient space.
= Discrete space
{parent=Topology}
{wiki=Discrete_space}
A topological space is discrete when every subset is an <open set>.
= Connected-space zero-product dichotomy
{parent=Topology}
If continuous functions $f,g$ on a connected space satisfy $fg=0$ and never vanish simultaneously, then the disjoint open sets $\{f\ne0\}$ and $\{g\ne0\}$ cover the space. One is empty, so one function vanishes identically.
= Normal topological space
{parent=Topology}
{wiki=Normal_space}
A topological space is normal when every two disjoint closed subsets have disjoint open neighbourhoods.
= Urysohn lemma
{c}
{parent=Normal topological space}
{wiki=Urysohn%27s_lemma}
If $A,B$ are disjoint closed subsets of a normal space, there is a continuous function $f:X\to[0,1]$ with $f|_A=0$ and $f|_B=1$.
= Tietze extension theorem
{c}
{parent=Urysohn lemma}
{wiki=Tietze_extension_theorem}
Every continuous real-valued function on a closed subspace of a normal space extends continuously to the whole space. If its values lie in a closed interval, the extension can be kept in that interval.
= Closed G-delta set as a zero set
{parent=Normal topological space}
A closed subset $A$ of a normal space is a countable intersection of open sets exactly when it is the zero set of some continuous function $f:X\to[0,1]$.
= Compact space
{parent=Topology}
{wiki=Compact_space}
A topological space is compact when every open cover has a finite subcover. Every closed subspace of a compact space is compact.
= Compact
{synonym}
= Compact set
{synonym}
= Continuous image of a compact space
{parent=Compact space}
The continuous image of a compact space is compact: pull an open cover of the image back to the domain, take a finite subcover there, and map it forward.
= Quotient of a compact space
{parent=Compact space}
Every quotient of a compact space is compact because the quotient projection is continuous and surjective.
= Lebesgue number lemma
{parent=Compact space}
{c}
{wiki}
Every open cover of a compact metric space has a number $\delta>0$ such that every subset of diameter less than $\delta$ lies in one member of the cover.
= Normality of a compact Hausdorff space
{parent=Compact space}
Every compact Hausdorff space is normal: disjoint closed subsets have disjoint open neighbourhoods. Pointwise Hausdorff separation becomes uniform first over one compact closed set and then over the other by taking finite subcovers.
= Locally compact space
{parent=Compact space}
{wiki=Locally_compact_space}
A space is locally compact when every point has a base of neighbourhoods containing compact neighbourhoods. A compact Hausdorff space is locally compact because regular separation produces $x\in V\subseteq\overline V\subseteq U$.
= Compactly detected closed-set theorem in a locally compact Hausdorff space
{parent=Locally compact space}
If $X$ is locally compact Hausdorff and $A\cap K$ is closed in every compact subset $K$, then $A$ is closed. Around each point outside $A$, use a compact neighbourhood and the relative openness of its complement of $A\cap K$.
= Hausdorff space
{parent=Topology}
{c}
{wiki}
A Hausdorff space separates any two distinct points by disjoint open neighborhoods.
= Compact subset of a Hausdorff space
{parent=Hausdorff space}
Every compact subset of a Hausdorff space is closed. For a point outside the compact set, separate it from each point of the set and use a finite subcover to obtain one neighbourhood disjoint from the whole set.
= Compact-to-Hausdorff continuous bijection theorem
{parent=Hausdorff space}
A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. It maps closed sets to compact sets, which are closed in the codomain, so it is a closed map and its inverse is continuous.
= Closed diagonal theorem
{parent=Hausdorff space}
{wiki}
A space is Hausdorff exactly when its diagonal is closed in its square.
= Homeomorphism
{parent=Topology}
{wiki}
A homeomorphism is a bijective continuous map whose inverse is continuous. Two spaces related by one have the same topological properties.
= Homeomorphic
{synonym}
= Closed map
{parent=Homeomorphism}
{wiki}
A closed map sends closed subsets of its domain to closed subsets of its codomain. A bijection is a homeomorphism exactly when it is continuous and closed.
= Unit sphere
{title2=$S^n$}
{parent=Topology}
{wiki=Unit_sphere}
The unit sphere in $\mathbb R^{n+1}$ is
$$
S^n=\{x\in\mathbb R^{n+1}:\|x\|=1\}.
$$
= Closed graph theorem for compact spaces
{parent=Topology}
{wiki}
A map between compact Hausdorff spaces is continuous exactly when its graph is closed.
= Closed-graph criterion with compact codomain
{parent=Closed graph theorem for compact spaces}
If $X,Y$ are metric spaces, $Y$ is compact, and the graph of $f:X\to Y$ is closed, then $f$ is continuous. Any failure of sequential continuity gives a subsequence whose images stay away from the proposed limit; compactness produces a convergent image subsequence, and closedness of the graph forces its limit to be the correct value.
= Topological surface
{parent=Topology}
{wiki}
= Surface quotient by a free finite action
{parent=Topological surface}
A free action of a finite group on a topological surface has a surface quotient: sufficiently small coordinate discs have disjoint translates and descend homeomorphically to quotient neighbourhoods.
= Jordan curve theorem
{parent=Topological surface}
{c}
{wiki}
Every simple closed curve on the sphere separates it into two connected components, each having the curve as its boundary.
A topological surface is a Hausdorff second-countable space locally homeomorphic to the plane.
= Polygonal schema
{parent=Topological surface}
{wiki}
A polygonal schema constructs a surface by identifying paired polygon edges.
= Polygonal-schema Euler count
{parent=Polygonal schema}
Identifying the $2n$ sides of one polygon in pairs gives a cell structure with one face, $n$ edges and some number $V$ of vertex classes. For a closed orientable surface of genus $g$,
$$
V-n+1=2-2g.
$$
Since $V\geq1$, every such schema satisfies $n\geq2g$.
= Regular hyperbolic octagon fundamental polygon
{parent=Polygonal schema}
A regular hyperbolic octagon with opposite sides paired gives a genus-two surface when each angle is $\pi/4$. The eight vertices then form one smooth quotient point because their angles sum to $2\pi$.
= Torus
{parent=Topological surface}
{wiki}
The torus is the product of two circles and can be formed by identifying opposite sides of a square.
= Fundamental group of the torus
{parent=Torus}
For the standard longitude $a$ and meridian $b$,
$$
\pi_1(T^2)\cong\langle a,b\mid[a,b]=1\rangle\cong\mathbb Z^2.
$$
= Möbius band
{c}
{parent=Topological surface}
{wiki}
The Möbius band deformation retracts onto its core circle. Under this retraction its boundary circle has degree two, so the boundary class is the square of the core class in the fundamental group.
= Klein bottle
{c}
{parent=Topological surface}
{wiki}
The Klein bottle is obtained from a square by identifying one pair of opposite sides with matching direction and the other pair with reversed direction.
= Orientation double cover of the Klein bottle
{parent=Klein bottle}
For
$$
(x,y)\simeq(x+c,(-1)^cy+d),
$$
the transformations with even $c$ form an index-two translation subgroup. Its quotient is a torus, giving the orientation double cover $T^2\to K$.
= Klein-bottle mapping-cylinder obstruction to simple connectivity
{parent=Orientation double cover of the Klein bottle}
Let $Y$ be the mapping cylinder of the orientation double cover $T^2\to K$ with its free end omitted. If a Hausdorff space contains $Y$ as an open subset, van Kampen extends the orientation quotient of $\pi_1(K)$ to a surjection of the ambient fundamental group onto $C_2$. The ambient space is therefore not simply connected.
= Flat Klein-bottle geodesic model
{parent=Klein bottle}
In a flat fundamental square, straight horizontal, vertical and rational-slope lines project to closed geodesics. The central horizontal glide axis is one-sided, a vertical line is two-sided, and the line $(2t,t)$ closes with one transverse self-intersection.
= Double cone singularity
{parent=Topological surface}
{wiki}
The vertex of a double cone is not a surface point because its punctured neighborhood disconnects into two pieces.
= Triangle KKM lemma
{parent=Topology}
{wiki=Knaster–Kuratowski–Mazurkiewicz_lemma}
If three closed sets cover a triangle and respectively contain its three opposite sides, then their triple intersection is nonempty.
= Distance-function barycentric map
{parent=Triangle KKM lemma}
Normalized distances to three closed sets give continuous barycentric coordinates; a covering forces at least one coordinate to vanish, placing the image on the simplex boundary.
= No-retraction covering principle
{parent=Triangle KKM lemma}
A closed-cover intersection statement for a simplex is equivalent to the nonexistence of a boundary-valued map that preserves every face.
= Hyperbolic geometry
{parent=Geometry and topology}
{wiki}
Hyperbolic geometry studies spaces of constant negative curvature.
= Hyperbolic plane
{parent=Hyperbolic geometry}
{wiki}
The hyperbolic plane is the simply connected two-dimensional geometry of constant curvature minus one.
= Finite-order elliptic transformation
{parent=Hyperbolic plane}
{wiki}
A noncentral finite-order orientation-preserving hyperbolic isometry fixes an interior point and is elliptic.
= Poincare disc model
{parent=Hyperbolic geometry}
{c}
{wiki=Poincaré_disk_model}
The Poincare disc carries the metric $4|dz|^2/(1-|z|^2)^2$ on $|z|<1$.
= Area of a hyperbolic disc
{parent=Poincare disc model}
In curvature minus one, a hyperbolic disc of radius $R$ has area
$$
A(R)=4\pi\sinh^2(R/2)=2\pi(\cosh R-1).
$$
In the Poincare disc this follows by integrating $4r(1-r^2)^{-2}\,dr\,d\theta$ up to Euclidean radius $\tanh(R/2)$.
= Radial chain of equal hyperbolic discs
{parent=Area of a hyperbolic disc}
Hyperbolic discs of area $\pi/2$ have radius $\log2$. Starting at the origin and placing their centers successively along one radial geodesic gives center distance $n\log4$ and Euclidean coordinate
$$
r_n=\frac{4^n-1}{4^n+1}.
$$
= Geodesic obstruction for a horizontal chain of hyperbolic discs
{parent=Poincare disc model}
An isometry sends the centers of a collinear family of equal hyperbolic discs to points on one hyperbolic geodesic. In the upper-half-plane model no geodesic contains three distinct points on a horizontal Euclidean line, since geodesics are vertical lines or semicircles orthogonal to the boundary.
= Hyperbolic length in the Poincare disc
{parent=Poincare disc model}
{c}
For a piecewise smooth curve $z(t)$ in the Poincare disc, its hyperbolic length is
$$
L(z)=\int\frac{2|\dot z(t)|}{1-|z(t)|^2}\,dt.
$$
= Hyperbolic length
{synonym}
= Geodesics of the Poincare disc
{parent=Poincare disc model}
{c}
The geodesics are diameters and arcs of circles orthogonal to the unit circle. A radial segment is length minimizing because
$$
\int\frac{2\sqrt{\dot r^{,2}+r^2\dot\theta^{,2}}}{1-r^2}\,dt
\geq2\operatorname{artanh}r,
$$
with equality for constant angle and monotone radius. Hence the distance from the origin to $z$ is $2\operatorname{artanh}|z|$.
= Euclidean circle representing a perpendicular hyperbolic line
{parent=Poincare disc model}
{c}
If $P$ is at hyperbolic distance $\rho$ from the origin, the Poincare-disc geodesic through $P$ perpendicular to $OP$ is represented by a Euclidean circle of radius $\operatorname{csch}\rho$ whose centre is at Euclidean distance $\coth\rho$ from the origin.
= Poincare half-plane model
{parent=Hyperbolic geometry}
{c}
{wiki=Poincaré_half-plane_model}
The upper half-plane carries the hyperbolic metric $(dx^2+dy^2)/y^2$.
= Geodesic in the Poincare half-plane model
{parent=Poincare half-plane model}
{c}
{wiki=Poincaré_half-plane_model#Geodesics}
The geodesics in the upper half-plane are vertical lines and semicircles centred on the boundary axis. Indeed, the length integrand $\sqrt{1+y'^2}/y$ has the <Beltrami identity>
$$
\frac1{y\sqrt{1+y'^2}}=\frac1R,
$$
which integrates to $(x-c)^2+y^2=R^2$. Swapping the coordinate axes gives circles centred on the vertical boundary for the metric $(dx^2+dy^2)/x^2$.
= PSL2(R)
{title2=$PSL_2(\mathbb R)$}
{parent=Poincare half-plane model}
{c}
{wiki=Projective_linear_group}
PSL2(R) is $SL_2(\mathbb R)/\{\pm I\}$. It acts on the upper half-plane by the Möbius transformations
$$
z\longmapsto\frac{az+b}{cz+d}
$$
and is the group of orientation-preserving hyperbolic isometries.
= Hyperbolic reflection
{parent=Poincare half-plane model}
Reflection in a hyperbolic line is the unique orientation-reversing hyperbolic isometry fixing that line pointwise. The product of reflections in intersecting lines is a rotation through twice their angle.
= Hyperbolic half-turn
{parent=Poincare half-plane model}
A hyperbolic half-turn is the unique orientation-preserving isometry of order two fixing a given point. It exchanges every pair of points lying at equal opposite distances along a geodesic through its centre.
= Cayley transform
{parent=Hyperbolic geometry}
{c}
{wiki}
The map $z\mapsto i(1+z)/(1-z)$ is an isometry from the Poincare disc to the upper half-plane.
= Hyperboloid model
{parent=Hyperbolic geometry}
{wiki}
The hyperboloid model of the curvature-minus-one hyperbolic plane is
$$
\{x\in\mathbb R^3:-x_0^2+x_1^2+x_2^2=-1, x_0>0\}.
$$
Its metric is the restriction of the Minkowski bilinear form to tangent planes, and its geodesics are intersections with two-dimensional linear subspaces through the origin.
= Minkowski inner product
{parent=Hyperboloid model}
{c}
{wiki=Minkowski_space}
On $\mathbb R^{n+1}$ the Minkowski inner product with mostly-positive convention is
$$
\langle x,y\rangle_M=-x_0y_0+x_1y_1+\cdots+x_ny_n.
$$
= Lambert quadrilateral
{parent=Hyperbolic geometry}
{c}
{wiki}
A Lambert quadrilateral is a hyperbolic quadrilateral with three right angles. If its remaining angle is $\theta$ and the two sides incident with that angle have lengths $a,b$, then
$$
\cos\theta=\tanh a\tanh b.
$$
= Lambert quadrilateral identity
{synonym}
= Hyperbolic triangle
{parent=Hyperbolic geometry}
{wiki}
A hyperbolic triangle is bounded by three hyperbolic geodesics and may have vertices on the ideal boundary.
= Ideal vertex
{parent=Hyperbolic triangle}
{wiki=Ideal_triangle}
An ideal vertex lies on the boundary at infinity and has internal angle zero.
= Hyperbolic triangle area
{parent=Hyperbolic triangle}
For curvature $-1$, a triangle of angles $\alpha,\beta,\gamma$ has area $\pi-\alpha-\beta-\gamma$ by Gauss--Bonnet.
= Hyperbolic polygon area
{parent=Hyperbolic triangle area}
For a geodesic hyperbolic $n$-gon of curvature $-1$ with interior angles $\alpha_1,\ldots,\alpha_n$, triangulation or the Gauss--Bonnet theorem gives
$$
\operatorname{area}=(n-2)\pi-\sum_{j=1}^n\alpha_j.
$$
= Hyperbolic law of cosines
{parent=Hyperbolic triangle}
{wiki=Hyperbolic_law_of_cosines}
For opposite side $a$, $\cosh a=(\cos\alpha+\cos\beta\cos\gamma)/(\sin\beta\sin\gamma)$.
= Hyperbolic law of sines
{parent=Hyperbolic geometry}
{wiki=Hyperbolic_law_of_sines}
For a hyperbolic triangle, $\sinh a/\sin\alpha=\sinh b/\sin\beta=\sinh c/\sin\gamma$.