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# 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 S2. Namely, given a surface S lying in R3, the Gauss map is a continuous map $N:S\to S^2$ such that N(p) 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 $\mathbb{R}^n$, this way we get a map from a hypersurface to the unit sphere $S^{n-1}\in \mathbb{R}^n$.

For a general oriented k-submanifold of $\mathbb{R}^n$ the Gauss map can be also be defined, and its target space is the oriented Grassmannian $\tilde{G}_{k,n}$, i.e. the set of all oriented k-planes in $\mathbb{R}^n$. 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 S of dimension k in an oriented ambient Riemannian manifold M of dimension n. In that case, the Gauss map then goes from S 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.

03-10-2013 05:06:04