# Shape operator on a hypersurface

## Definition

### For a hypersurface in any dimension

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

$v \mapsto - \nabla_vN$

where $\nabla_vN$ is the component of $D_vN$ (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 $\Gamma(T^*M) \otimes \Gamma(TM)$.

### 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 $\R^3$.