Gauss map

English

A Gauss map matches each point on the surface (left) with the point on the unit sphere (right) representing the orthogonal vector at said point.

Etymology

Named after German mathematician Template:W.

Noun

1. Template:Lb A map from a given oriented surface in Euclidean space to the unit sphere which maps each point on the surface to a unit vector orthogonal to the surface at that point. Template:C
   • 1969 [Van Nostrand], Template:W, A Survey of Minimal Surfaces, 2014, Dover, Unabridged republication, page 73,

     There exist complete generalized minimal surfaces, not lying in a plane, whose Gauss map lies in an arbitrarily small neighborhood on the sphere.

   • 1985, R. G. Burns (translator), B. A. Dubrovin, Template:W, Template:W, Modern Geometry― Methods and Applications: Part II: The Geometry and Topology of Manifolds, Springer, Template:W, page 114,

     14.2.2 Theorem The integral of the Gaussian curvature over a closed hypersurface in Euclidean $n$-space is equal to the degree of the Gauss map of the surface, multiplied by ${\gamma }_n$ (the Euclidean volume of the unit $(n-1)$-sphere).

   • Template:Quote-book

See also

• Template:L

Further reading

• Gauss Map on Template:W