Gauss-Weingarten map

From Diffgeom
Jump to: navigation, search

Definition

Suppose M is a differential manifold of dimension m, embedded smoothly inside \R^n.

For non-oriented submanifolds

If M is not assumed to have an orientation, the Gauss-Weingarten map is a map from M to the Grassmannian manifold of m-dimensional subspaces of \R^n, as follows: any point p \in M is mapped to the vector subspace of \R^n parallel to the tangent space T_pM.

For oriented submanifolds

If we give an orientation to M the Gauss-Weingarten map is a map from M to the oriented Grassmannian manifold of m-dimensional subspaces of \R^n as follows: any point p \in M is sent to the vector subspace parallel to the tangent space T_p M, equipped with the orientation.

For codimension one submanifolds

For codimension one oriented submanifolds, we can identify the oriented Grassmannian with the sphere S^{n-1} = S^m, whereas for codimension one non-oriented submanifolds, we can identify the Grassmannian with real projective space \R\mathbb{P}^m.

Tangent space can be replaced by normal space

We can define the Gauss-Weingarten map, instead, by sending each point to its normal space (the orthogonal complement of the tangent space to M in the tangent space to \R^n) with suitable orientation, if the manifold is oriented. The two maps are equivalent, under the isomorphism between the Grassmannian of m-dimensional subspaces, and the Grassmannian of (n-m)-dimensional subspaces.