Shape operator on a hypersurface

Definition

For a hypersurface in any dimension

Suppose $M$ is a $n$-dimensional manifold embedded inside ${\mathbb{R}}^{n+1}$. The shape operator on $M$ associates, to every point $p\in M$, a linear map from $T_{p}M$ to $T_{p}M$, given by:

$v\mapsto -{\mathrm{\nabla }}_{v}N$

where ${\mathrm{\nabla }}_{v}N$ is the component of $D_{v}N$ (the covariant derivative of the normal in terms of $v$) in the $T_p(M)$-direction.

Equivalently, the shape operator is the differential of the Gauss map for the hypersurface $M$, namely the map:

$M\to S^n$

that sends a point in $M$ to the normal direction at that point.

The shape operator can be viewed as a section of the bundle $\mathrm{\Gamma }(T^{*}M)\otimes \mathrm{\Gamma }(TM)$.

For a regular surface in 3-space

This is the special case of the above, in the situation where $M$ is a regular surface inside ${\mathbb{R}}^3$.