Gauss-Weingarten map
Contents
Definition
Suppose is a differential manifold of dimension
, embedded smoothly inside
.
For non-oriented submanifolds
If is not assumed to have an orientation, the Gauss-Weingarten map is a map from
to the Grassmannian manifold of
-dimensional subspaces of
, as follows: any point
is mapped to the vector subspace of
parallel to the tangent space
.
For oriented submanifolds
If we give an orientation to the Gauss-Weingarten map is a map from
to the oriented Grassmannian manifold of
-dimensional subspaces of
as follows: any point
is sent to the vector subspace parallel to the tangent space
, equipped with the orientation.
For codimension one submanifolds
For codimension one oriented submanifolds, we can identify the oriented Grassmannian with the sphere , whereas for codimension one non-oriented submanifolds, we can identify the Grassmannian with real projective space
.
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 in the tangent space to
) with suitable orientation, if the manifold is oriented. The two maps are equivalent, under the isomorphism between the Grassmannian of
-dimensional subspaces, and the Grassmannian of
-dimensional subspaces.