Second Fundamental Form - Hypersurface in A Riemannian Manifold

Hypersurface in A Riemannian Manifold

In Euclidean space, the second fundamental form is given by

where is the Gauss map, and the differential of regarded as a vector valued differential form, and the brackets denote the metric tensor of Euclidean space.

More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by ) of a hypersurface,

where denotes the covariant derivative of the ambient manifold and a field of normal vectors on the hypersurface. (If the affine connection is torsion-free, then the second fundamental form is symmetric.)

The sign of the second fundamental form depends on the choice of direction of (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).