Second fundamental form

Category:Riemannian geometry 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,

\nwhere denoted covariant derivative and n a field of normal vectors on hypersurface.\nThe sign of second fundamental form depends on the choice of direction of n (which is the same as choice of orientation on the hypersurface).

\nThe second fundamental form can be generalized to arbitrary codimension. \nIn that case it is a quadratic form on the tangent space with values in the normal space \nand it can be defined by

where denotes normal projection of covariant derivative . \nIn Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

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 of N with induced metric can be expressed\nusing second fundamental form and , the curvature tensor of M: