Shape operator on a hypersurface: Difference between revisions

Revision as of 14:17, 6 April 2008

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 $T_{p} M$ to $T_{p} M$ , given by:

$v \mapsto - \nabla_{v} N$

where $\nabla_{v} N$ 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 $Γ (T^{*} M) \otimes Γ (T M)$ .

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 $R^{3}$ .

@@ Line 5: / Line 5: @@
 Suppose <math>M</math> is a <math>n</math>-dimensional manifold embedded inside <math>\R^{n+1}</math>. The '''shape operator''' on <math>M</math> associates, to every point <math>p \in M</math>, a linear map from <math>T_pM</math> to <math>T_pM</math>, given by:
-<math>v \mapsto - \nabla_vN</math>
+{{quotation|<math>v \mapsto - \nabla_vN</math>}}
 where <math>\nabla_vN</math> is the component of <math>D_vN</math> (the covariant derivative of the normal in terms of <math>v</math>) in the <math>T_p(M)</math>-direction.
+Equivalently, the shape operator is the differential of the [[Gauss map]] for the hypersurface <math>M</math>, namely the map:
+<math>M \to S^n</math>
+that sends a point in <math>M</math> to the normal direction at that point.
 The shape operator can be viewed as a section of the bundle <math>\Gamma(T^*M) \otimes \Gamma(TM)</math>.