Second fundamental form - Definition

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,

<math>I\!I(v,w)=\langle S(v),w\rangle= -\langle \nabla_v n,w\rangle=\langle n,\nabla_v w\rangle,<math>

where <math>\nabla_v w<math> 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

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

where <math>\nabla^\bot_v w <math> denotes normal projection of covariant derivative <math>\nabla_v w <math>. In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

<math>\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.<math>

For general Riemannian manifold one has to add the curvature of ambient space, if N is a manifold embeded in a Riemannian manifold (M,g) then the curvature tensor <math>R_N <math> of N with induced metric can be expressed using second fundamental form and <math>R_M <math>, the curvature tensor of M:

<math>\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.<math>