# Gauss map

In differential geometry, the Gauss map (named, like so many things, after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere [itex]S^2[itex]. Namely, given a surface [itex]S[itex] lying in R3, the Gauss map is a continuous map [itex]N:S\to S^2[itex] such that [itex]N(p)[itex] is orthogonal to S at p.

The Gauss map can be defined (globally) if and only if the surface is orientable, but it is always defined locally (i.e. on a small piece of the surface). The Jacobian of the Gauss map is equal to Gauss curvature, the differential of the Gauss map is called shape operator.

## Simplified explanation

Each point on the surface has a normal. That is, a vector orthogonal to the surface at that point. Now, move this vector to the origin. Do this for all such vectors on the surface. What we get is a surface on the sphere (possibly with overlaps). This is called the Gauss map. A similar concept in 2 dimensions with curves is the radial of a curve.

## Generalizations

The Gauss map can be defined the same way for hypersurfaces in [itex]\mathbb{R}^n[itex], this way we get a map from a hypersurface to the unit sphere [itex]S^{n-1}\in \mathbb{R}^n[itex].

For a general oriented k-submanifold of [itex]\mathbb{R}^n[itex] the Gauss map can be also be defined, and its target space is the oriented Grassmannian [itex]\tilde{G}_{k,n}[itex], i.e. the set of all oriented [itex]k[itex]-planes in [itex]\mathbb{R}^n[itex]. In this case a point on the submanifold is mapped to its oriented tangent subspace. It should be noted that in Euclidean 3-space, an oriented 2-plane is characterized by a normal unit normal vector, hence this is consistent with the definition above.

Finally, the notion of Gauss map can be generalized to an oriented submanifold [itex]S[itex] of dimension [itex]k[itex] in an oriented ambient Riemannian manifold [itex]M[itex] of dimension [itex]n[itex]. In that case, the Gauss map then goes from [itex]S[itex] to the set of tangent [itex]k[itex]-planes in the tangent bundle [itex]TM[itex]. The target space for the Gauss map [itex]N[itex] is a Grassmann bundle built on the tangent bundle [itex]TM[itex].

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