Extrinsic curvature

Basic approach

In the case of a surface, in @needham2021visual the author defines it as the local area magnification factor of the Gauss map. It is Gaussian curvature#Definition 2.

In page 134 he shows that if we draw a quadrilateral following principal directions then the area magnification is

the product of the principal curvatures. It is the determinant of the shape operator or the Weingarten map.

Generalization to hypersurfaces

See:

• Spivak approach with the covariant derivative, and
• the moving frames approach.