= Second fundamental form
{title2=$II$}
{wiki}
For an <embedded submanifold> $M\subseteq N$ of a <Riemannian manifold>, the vector-valued second fundamental form is
$$
A(X,Y)=(\overline\nabla_XY)^\perp.
$$
Here $\overline\nabla$ is the ambient <Levi-Civita connection> and the superscript denotes projection onto the <normal bundle>. For a hypersurface in Euclidean space with unit normal $N$, the scalar form is $II(X,Y)=\langle A(X,Y),N\rangle=\langle-dN(X),Y\rangle$. In a surface parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.
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