Second fundamental form

From Wikipedia, the free encyclopedia

In differential geometry, the second fundamental form is a quadratic form on the tangent space of a hypersurface, usually denoted by II. It is an equivalent way to describe the shape operator (denoted by S) of a hypersurface,

I\!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 the same as choice of orientation on the hypersurface).

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 space and it can be defined by

I\!I(v,w)=\nabla^\bot_v w,

where \nabla^\bot_v w denotes normal projection of covariant derivative \nabla_v w.

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

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

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 RN of N with induced metric can be expressed using second fundamental form and RM, the curvature tensor of M:

\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle I\!I(u,z),I\!I(v,w)\rangle-\langle I\!I(u,w),I\!I(v,z)\rangle.
Retrieved from "http://en.wikipedia.org../../../s/e/c/Second_fundamental_form.html"