Gauss map

== English == === Etymology === Named after German mathematician Carl Friedrich Gauss. === Noun === Gauss map (plural Gauss maps) (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 {\displaystyle n} -space is equal to the degree of the Gauss map of the surface, multiplied by γ n {\displaystyle \gamma _{n}} (the Euclidean volume of the unit ( n − 1 ) {\displaystyle (n-1)} -sphere). === See also === shape operator === Further reading === Gauss Map on Wolfram MathWorld