Generalizations
The Gauss map can be defined for hypersurfaces in Rn as a map from a hypersurface to the unit sphere Sn − 1 ∈ Rn.
For a general oriented k-submanifold of Rn the Gauss map can be also be defined, and its target space is the oriented Grassmannian, i.e. the set of all oriented k-planes in Rn. 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 via orthogonal complement. In Euclidean 3-space, this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as ), 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, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.
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