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.