Editing Gauss map (section)

==Generalizations==
{{Unreferenced section|date=May 2020}}
The Gauss map can be defined for [[hypersurface]]s in '''R'''<sup>''n''</sup> as a map from a hypersurface to the unit sphere ''S''<sup>''n'' &minus; 1</sup> &nbsp;⊆&nbsp; '''R'''<sup>''n''</sup>.

For a general oriented ''k''-[[submanifold]] of '''R'''<sup>''n''</sup> the Gauss map can also be defined, and its target space is the ''oriented'' [[Grassmannian]] 
<math>\tilde{G}_{k,n}</math>, i.e. the set of all oriented ''k''-planes in '''R'''<sup>''n''</sup>. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its oriented ''normal'' subspace; these are equivalent as <math>\tilde{G}_{k,n} \cong \tilde{G}_{n-k,n}</math> via orthogonal complement.
In [[Euclidean space|Euclidean 3-space]], this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as <math>\tilde{G}_{1,n} \cong S^{n-1}</math>), hence this is consistent with the definition above.

Finally, the notion of Gauss map can be generalized to an oriented submanifold ''X'' of dimension ''k'' in an oriented ambient [[Riemannian manifold]] ''M'' of dimension ''n''. In that case, the Gauss map then goes from ''X'' to the set of tangent ''k''-planes in the [[tangent bundle]] ''TM''. The target space for the Gauss map ''N'' is a [[Grassmann bundle]] built on the tangent bundle ''TM''. In the case where <math>M=\mathbf{R}^n</math>, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.