Gauss-Weingarten map

Definition

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

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 ${\mathbb{R}}^n$, as follows: any point $p\in M$ is mapped to the vector subspace of ${\mathbb{R}}^n$ parallel to the tangent space $T_{p}M$.

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 ${\mathbb{R}}^n$ 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 $\mathbb{R}{\mathbb{P}}^m$.

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 ${\mathbb{R}}^n$) 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.