Second fundamental form

Basic approach

It is a (0,2)-tensor defined in a Riemannian manifold of dimension 2 (surface) that reflect data about how is immersed in ${\mathbb{R}}^3$. It is obtained from the shape operator, by lowering indices.

Generalization with $\mathrm{\nabla }$ approach

See Gauss' Equation#For hypersurfaces.

Generalization to surfaces in 3-dimensional manifolds via moving frames

See geometry of a submanifold via moving frames.