Second fundamental form

In differential geometry, the second fundamental form is a quadratic form, usually denoted by II, on the tangent space of a hypersurface in a Riemannian manifold, such as a surface in three dimensional Euclidean space. It is an equivalent way to describe the shape operator (denoted by S) of a hypersurface,

\mathrm {I} \!\mathrm {I} (v,w)=\langle S(v),w\rangle =-\langle \nabla _{v}n,w\rangle =\langle n,\nabla _{v}w\rangle ,

where $\nabla _{v}w$ denoted covariant derivative and n a field of normal vectors on hypersurface. The sign of second fundamental form depends on the choice of direction of n (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).

The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by

\mathrm {I} \!\mathrm {I} (v,w)=(\nabla _{v}w)^{\bot },

where $(\nabla _{v}w)^{\bot }$ denotes the orthogonal projection of covariant derivative $\nabla _{v}w$ onto the normal bundle.

In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

\langle R(u,v)w,z\rangle =\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle .

This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium.

For general Riemannian manifold one has to add the curvature of ambient space, if N is a manifold embedded in a Riemannian manifold (M,g) then the curvature tensor $R_{N}$ of N with induced metric can be expressed using second fundamental form and $R_{M}$ , the curvature tensor of M: