Shape operator on a hypersurface

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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.