Gauss map

In differential geometry, the Gauss map forms (named after Carl F. Gauss ) a surface in Euclidean space to the unit sphere from.

Gauss wrote the first time in 1825 on the subject and published it in 1827.

Definition

Based on a given surface, which is in the Gaussian image so that an orthonormal to the surface of a unit vector is a continuous mapping, wherein, namely of the normal vector at is.

Properties

The Gauss map can be global, ie, if and only defined for all, if the surface is orientable. Local, that is, on a small piece of the surface, it can always be defined. The Jacobian of the Gauss map is equal to the Gaussian curvature, and the differential of the Gauss map is called the Weingarten map or shape operator.

Generalization

Analogous to the above definition, the Gauss map for n-dimensional oriented hypersurfaces in be defined.