Gauss map

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In differential geometry, the Gauss map (named, like so many things, 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 orthogonal to X at p.

The Gauss map can be defined (globally) if and only if the surface is orientable, but it is always defined locally (i.e. on a small piece of the surface). The Jacobian 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.

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Simplified explanation

Each point on the surface has a normal vector; that is, a vector orthogonal to the surface at that point. Now, move this vector to the origin. Do this for all such vectors on the surface. What we get is a surface on the 2-sphere (possibly with overlaps). This is called the Gauss map. A similar concept in 2 dimensions with curves is the radial of a curve.

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Generalizations

The Gauss map can be defined the same way for hypersurfaces in Rn, this way we get a map from a hypersurface to the unit sphere Sn − 1 ∈ Rn.

For a general oriented k-submanifold of Rn the Gauss map can be also be defined, and its target space is the oriented Grassmannian <math>\tilde{G}_{k,n}</math>, 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. It should be noted that in Euclidean 3-space, an oriented 2-plane is characterized by a unit normal vector, 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.

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Cusps of the Gauss Map

The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along a parabolic line) the Gauss map will have a Catastrophe theory#Fold catastrophe. This fold may contain cusps and these cusps were studied in depth by Thomas Banchoff, Terence Gaffney and Clint McCrory. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface. Cusps occur when:

  1. The surface has a bi-tangent plane
  2. A ridge crosses a parabolic line
  3. at the closure of the set of inflection points of the asymptotic curves of the surface.

There are two types of cusp elliptic cusp and hyperbolic cusps.


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References

  • Gauss, K. F., Disquisitiones generales circa superficies curvas (1827)
  • Gauss, K. F., General investigations of curves surfaces, English translation. Hewlett, New York: Raven Press (1965).
  • Banchoff, T., Gaffney T., McCrory C., Cusps of the Gauss Map, (1982) Research Notes in Mathematics 55, Pitman, London. online version
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