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 Carl Friedrich Gauss.

Noun

Gauss map (plural Gauss maps)

1. (geometry, differential geometry) 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.
   • 1969 [Van Nostrand], Robert Osserman, 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, A. T. Fomenko, S. P. Novikov, Modern Geometry― Methods and Applications: Part II: The Geometry and Topology of Manifolds, Springer, Graduate Texts in Mathematics, 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).

   • 2005, F. L. Zak, Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 5:

     In §2 we use results of §1 for the study of Gauss maps of projective varieties. The classical Gauss map associates to each point of a nonsingular real affine hypersurface the unit vector of the external normal at this point.

See also

• shape operator

Further reading

• Gauss Map on Wolfram MathWorld