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).
Read more about this topic: Second Fundamental Form
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