Gauss Map

The Gauss map is a function from an Orientable Surface in Euclidean Space to a Sphere. It associates to every point on the surface its oriented Normal Vector. For surfaces in 3-space, the Gauss map of the surface has Degree given by half the Euler Characteristic of the surface

$$\int\!\!\!\int _M K \,dA = 2\pi \chi(M) - \sum \alpha_i - \int_{\partial M} \kappa_g\, ds,$$

which works only for Orientable Surfaces.

See also Curvature, Nirenberg's Conjecture, Patch

References

Gray, A. ``The Local Gauss Map'' and ``The Gauss Map via Mathematica.'' §10.3 and §15.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 193-194 and 310-316, 1993.