Second Fundamental Form

Definition

Given $\sigma (u,v)\in {\mathbb{R}}^3$. Define $\mathbf{n}:=\frac{{\sigma }_u\times {\sigma }_v}{||{\sigma }_u\times {\sigma }_v||}$ $\mathrm{II}:=Ld{u}^2+2Mdudv+Nd{v}^2=-({\sigma }_{u}du+{\sigma }_{v}dv)\cdot ({\mathbf{n}}_{u}du+{\mathbf{n}}_{v}dv)=-d\sigma \cdot d\mathbf{n}$ Here, $L=-{\sigma }_u\cdot {\mathbf{n}}_u$ $2M=-({\sigma }_u\cdot {\mathbf{n}}_v+{\sigma }_v\cdot {\mathbf{n}}_u)$ $N=-{\sigma }_v\cdot {\mathbf{n}}_{v}d{v}^2$ or $L={\sigma }_{uu}\cdot \mathbf{n}$, $M={\sigma }_{uv}\cdot \mathbf{n}$, $N={\sigma }_{vv}\cdot \mathbf{n}$

Facts

$d\sigma ,dN$ are defined by Differential 1-Form