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Concept
Gauss map
Summary
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 a 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 Sn − 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
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 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 ), hence this is consistent with the definition above.
Finally, the notion of Gauss map can be generalized to an oriented submanifold X of dimension k in an oriented ambient Riemannian manifold M of dimension n. In that case, the Gauss map then goes from X to the set of tangent k-planes in the tangent bundle TM. The target space for the Gauss map N is a Grassmann bundle built on the tangent bundle TM. In the case where , the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.
The area of the image of the Gauss map is called the total curvature and is equivalent to the surface integral of the Gaussian curvature.
Official source
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