Editing Gauss map

{{Short description|Differential geometry topic}}
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{{More 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 this example, the curvature of a 2D-surface is mapped onto a 1D unit circle.]]

In [[differential geometry]], the '''Gauss map'''  of a [[Surface (topology)|surface]] is a [[function (mathematics)|function]] that maps each point in the surface to its [[normal direction]], a [[unit vector]] that is [[orthogonal]] to the surface at that point. Namely, given a surface ''X'' in [[Euclidean space]] '''R'''<sup>3</sup>, the Gauss map is a map ''N'': ''X'' → ''S''<sup>2</sup> (where ''S''<sup>2</sup> is the [[unit sphere]]) such that  for each ''p'' in ''X'', the function value ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p''. The Gauss map is named after [[Carl Friedrich Gauss|Carl F. Gauss]].

The Gauss map can be defined (globally) if and only if the surface is [[orientable]], in which case its [[Degree of a continuous mapping|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.<ref>{{Cite book |author1=Gauss, Karl Friedrich |title=General Investigations of Curved Surfaces of 1827 and 1825 |publisher=The Princeton University Library |year=1902 |translator1=Morehead, James Caddall |translator2=Hiltebeitel, Adam Miller}}</ref>{{Fact|date=November 2024}}

There is also a Gauss map for a [[Link (knot theory)|link]], which computes [[linking number]].

==Generalizations==
{{Unreferenced section|date=May 2020}}
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> &nbsp;⊆&nbsp; '''R'''<sup>''n''</sup>.

For a general oriented ''k''-[[submanifold]] of '''R'''<sup>''n''</sup> the Gauss map can 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. 

<math> \iint_R \pm|N_u \times N_v| \ du\, dv = \iint_R K|X_u \times X_v| \ du\, dv = \iint_S K \ dA</math>

The [[Gauss–Bonnet theorem]] links total curvature of a surface to its [[topology|topological]] properties.
==Cusps of the Gauss map==
[[File:Cusp of the Gauss map.png|thumb|A surface with a parabolic line and its Gauss map. A ridge passes through the parabolic line giving rise to a cusp on 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]].<ref>{{Cite journal |author1=McCrory, Clint |author2=Shifrin, Theodore |year=1984 |title=Cusps of the projective Gauss map |journal=Journal of Differential Geometry |volume=19 |pages=257–276|doi=10.4310/JDG/1214438432 |s2cid=118784720 }}</ref> 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==
{{Reflist}}
*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 Gauss Mappings'', (1982) Research Notes in Mathematics 55, Pitman, London. [http://www.math.brown.edu/~dan/cgm/index.html online version] {{Webarchive|url=https://web.archive.org/web/20080802155134/http://www.math.brown.edu/~dan/cgm/index.html |date=2008-08-02 }} <--broken link;  [https://www.emis.de/monographs/CGM/index.html Dan Dreibelbis' online version] (accessed 2023-07-01), {{Webarchive|url=https://web.archive.org/web/20080802155134/http://www.math.brown.edu/~dan/cgm/index.html |date=2008-08-02 }}
*Koenderink, J. J., ''Solid Shape'', MIT Press (1990)

== External links ==
* {{MathWorld | urlname=GaussMap | title=Gauss Map}}
* {{cite book|author1=Thomas Banchoff|author2=Terence Gaffney|author3=Clint McCrory|author4=Daniel Dreibelbis|title=Cusps of Gauss Mappings|series=Research Notes in Mathematics|volume=55|date=1982|publisher=Pitman Publisher Ltd.|location=London|isbn=0-273-08536-0|url=http://www.emis.de/monographs/CGM/index.html|accessdate=4 March 2016}}

{{Carl Friedrich Gauss}}

[[Category:Differential geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Riemannian geometry]]
[[Category:Surfaces]]
[[Category:Carl Friedrich Gauss]]