The Gauss map is a function N from an oriented surface M in Euclidean space R^3 to the unit sphere in R^3. It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point p on M is parallel to the tangent space at its image point on the sphere, the differential d N can be considered as a map of the tangent space at p into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.
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