Exponential map (${\exp }_p$)

For $p$ in a Riemannian manifold and $v$ sufficiently close to zero in $T_{p}M$, let ${\gamma }_v$ be the geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map isEquivalently, ${\exp }_p(tv)={\gamma }_v(t)$ wherever both sides are defined.