OurBigBook.comSite Source code

Euclidean invariance of the shape operator

For a proper Euclidean motion of Euclidean three-space

E (x) = A x + b

, transport the unit normal by

N (E (p)) = A N (p)

. The corresponding shape operators are orthogonally conjugate:

S_{E (p)} = A S_{p} A^{- 1} .

Their determinant and trace, hence Gaussian curvature and mean curvature, are unchanged.

 Ancestors (8)

 Incoming links (1)

Solution