 Second fundamental form

For an embedded submanifold $M\subseteq {\mathbb{R}}^{n+m}$, the vector-valued second fundamental form isFor an oriented hypersurface with unit normal $N$, the scalar form is $II(X,Y)=⟨A(X,Y),N⟩=⟨-dN(X),Y⟩$. In a surface parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.

The second fundamental form is a mathematical object used in differential geometry that provides a way to describe how a surface bends in a higher-dimensional space. Specifically, it is associated with a surface \( S \) embedded in a higher-dimensional Euclidean space, such as \(\mathbb{R}^3\).