In differential geometry, the Gauss map (named after Carl Friedrich Gauss|Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N(p) is a unit vector orthogonal to X at p, namely the normal vector to X at p.
The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian matrix and determinant|Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential (calculus)|differential of the Gauss map is called the shape operator.
Gauss first wrote a draft on the topic in 1825 and published in 1827.
There is also a Gauss map for a Link (knot theory)|link, which computes linking number.