# shape operator

The shape operator $S$ of a surface $\Sigma$ in ${\mathbb{R}}^{3}$ is the derivative of the sphere map $N:\Sigma\to S^{2}$ given by $N(p)=$ unit normal vector field at $p$. So at each $p$, $S(p)=d_{p}N$ and it is the linear transformation $S(p):T_{p}\Sigma\to T_{N(p)}S^{2}$. This is important, because the determinant defines the Gaussian curvature at $p$ in $\Sigma$.

Title shape operator ShapeOperator 2013-03-22 16:04:31 2013-03-22 16:04:31 juanman (12619) juanman (12619) 6 juanman (12619) Definition msc 53A05 SecondFundamentalForm