Editing Gauss map (section)

{{Short description|Differential geometry topic}}
{{About|differential geometry||}}
{{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]].