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Indifferential geometry, theGauss map of asurface is afunction that maps each point in the surface to aunit vector that isorthogonal to the surface at that point. Namely, given a surfaceX inEuclidean spaceR3, the Gauss map is a mapN:X →S2 (whereS2 is theunit sphere) such that for eachp inX, the function valueN(p) is a unit vector orthogonal toX atp. The Gauss map is named afterCarl F. Gauss.
The Gauss map can be defined (globally) if and only if the surface isorientable, in which case itsdegree is half theEuler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). TheJacobian determinant of the Gauss map is equal toGaussian curvature, and thedifferential of the Gauss map is called theshape operator.
Gauss first wrote a draft on the topic in 1825 and published in 1827.[1][citation needed]
There is also a Gauss map for alink, which computeslinking number.
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The Gauss map can be defined forhypersurfaces inRn as a map from a hypersurface to the unit sphereSn − 1 ⊆ Rn.
For a general orientedk-submanifold ofRn the Gauss map can also be defined, and its target space is theorientedGrassmannian, i.e. the set of all orientedk-planes inRn. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its orientednormal subspace; these are equivalent as via orthogonal complement.InEuclidean 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 submanifoldX of dimensionk in an oriented ambientRiemannian manifoldM of dimensionn. In that case, the Gauss map then goes fromX to the set of tangentk-planes in thetangent bundleTM. The target space for the Gauss mapN is aGrassmann bundle built on the tangent bundleTM. 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 thetotal curvature and is equivalent to thesurface integral of theGaussian curvature. This is the original interpretation given by Gauss.
TheGauss–Bonnet theorem links total curvature of a surface to itstopological properties.
The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along aparabolic line) the Gauss map will have afold catastrophe.[2] This fold may containcusps and these cusps were studied in depth byThomas Banchoff,Terence Gaffney andClint McCrory. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface. Cusps occur when:
There are two types of cusp:elliptic cusp andhyperbolic cusps.
