Gauss map

Named after German mathematician Carl Friedrich Gauss.

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 ${\displaystyle n}$-space is equal to the degree of the Gauss map of the surface, multiplied by ${\displaystyle \gamma _{n}}$ (the Euclidean volume of the unit ${\displaystyle (n-1)}$-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.

• shape operator

• Gauss Map on Wolfram MathWorld