Gauss map

In differential geometry, the Gauss map (named after 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 determinant of the Gauss map is equal to Gaussian curvature, and the 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, which computes linking number.

The Gauss map can be defined for hypersurfaces in Rⁿ as a map from a hypersurface to the unit sphere S^{n − 1} ∈ Rⁿ.

For a general oriented k-submanifold of Rⁿ the Gauss map can also be defined, and its target space is the oriented Grassmannian ${\displaystyle {\tilde {G}}_{k,n}}$, i.e. the set of all oriented k-planes in Rⁿ. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its oriented normal subspace; these are equivalent as ${\displaystyle {\tilde {G}}_{k,n}\cong {\tilde {G}}_{n-k,n}}$ via orthogonal complement. In Euclidean 3-space, this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as ${\displaystyle {\tilde {G}}_{1,n}\cong S^{n-1}}$), hence this is consistent with the definition above.

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