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{{no footnotes|date=July 2011}}
[[Image:Gauss map.svg|thumb|400px|The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere]]
In [[differential geometry]], the '''Gauss map''' (named after [[Carl Friedrich Gauss|Carl F. Gauss]]) maps a [[surface]] in [[Euclidean space]] '''R'''<sup>3</sup> to the unit [[sphere]] ''S''<sup>2</sup>. Namely, given a surface ''X'' lying in '''R'''<sup>3</sup>, the Gauss map is a continuous map ''N'': ''X'' → ''S''<sup>2</sup> 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 matrix and determinant|Jacobian]] determinant of the Gauss map is equal to [[Gaussian curvature]], and the [[differential (calculus)|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 (knot theory)|link]], which computes [[linking number]].
 
==Generalizations==
The Gauss map can be defined for [[hypersurface]]s in '''R'''<sup>''n''</sup> as a map from a hypersurface to the unit sphere ''S''<sup>''n'' &minus; 1</sup> ∈ '''R'''<sup>''n''</sup>.
 
For a general oriented ''k''-[[submanifold]] of '''R'''<sup>''n''</sup> 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 '''R'''<sup>''n''</sup>. 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 <math>\tilde{G}_{k,n} \cong \tilde{G}_{n-k,n}</math> via orthogonal complement.
In [[Euclidean space|Euclidean 3-space]], this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as <math>\tilde{G}_{1,n} \cong S^{n-1}</math>), 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 <math>M=\mathbf{R}^n</math>, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.
 
==Total curvature==
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]]. This is the original interpretation given by Gauss. The [[Gauss-Bonnet theorem]] links total curvature of a surface to its [[topology|topological]] properties.
:<math> \iint_R |N_u \times N_v| \ du\, dv = \iint_R K|X_u \times X_v| \ du\, dv = \iint_R K \ dA</math>
 
==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|fold catastrophe]]. This fold may contain [[cusp (singularity)|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:
#The surface has a bi-tangent plane
#A [[ridge (differential geometry)|ridge]] crosses a parabolic line
#at the closure of the set of inflection points of the [[asymptotic curve]]s of the surface.
There are two types of cusp: ''elliptic cusp'' and ''hyperbolic cusps''.
 
==References==
*Gauss, K. F., ''Disquisitiones generales circa superficies curvas'' (1827)
*Gauss, K. F., ''General investigations of curved 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. [http://www.math.brown.edu/~dan/cgm/index.html online version]
 
== External links ==
* {{MathWorld | urlname=GaussMap | title=Gauss Map}}
 
{{Chaos theory}}
 
[[Category:Differential geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Riemannian geometry]]
[[Category:Surfaces]]

Latest revision as of 02:34, 11 March 2014

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