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Gauss map

From Academic Kids

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

For a general oriented k-submanifold of <math>\mathbb{R}^n<math> the Gauss map can be also be defined, and its target space is the oriented Grassmannian <math>\tilde{G}_{k,n}<math>, i.e. the set of all oriented <math>k<math>-planes in <math>\mathbb{R}^n<math>. 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 <math>S<math> of dimension <math>k<math> in an oriented ambient Riemannian manifold <math>M<math> of dimension <math>n<math>. In that case, the Gauss map then goes from <math>S<math> to the set of tangent <math>k<math>-planes in the tangent bundle <math>TM<math>. The target space for the Gauss map <math>N<math> is a Grassmann bundle built on the tangent bundle <math>TM<math>.

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