en>Braddarb at 00:10, 24 January 2014

2014-01-24T00:10:54Z

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[[File:MorinSurfaceAsSphere'sInsideVersusOutside.PNG|thumb|A [[Morin surface]], an [[immersion (mathematics)|immersion]] used in [[sphere eversion]].]]
In [[mathematics]], more specifically in [[differential geometry]] and [[topology]], various types of [[Function (mathematics)|functions]] between [[manifold]]s are studied, both as objects in their own right and for the light they shed

== Types of maps ==
Just as there are various types of manifolds, there are various types of maps of manifolds.

[[File:PDIFF.svg|135px|thumb|[[PDIFF]] serves to relate DIFF and PL, and it is equivalent to PL.]]
In [[geometric topology]], the basic types of maps correspond to various [[category (mathematics)|categories]] of manifolds: DIFF for [[smooth function]]s between [[differentiable manifold]]s, PL for [[piecewise linear function]]s between [[piecewise linear manifold]]s, and TOP for [[continuous function]]s between [[topological manifold]]s. These are progressively weaker structures, properly connected via [[PDIFF]], the category of [[piecewise]]-smooth maps between piecewise-smooth manifolds.

In addition to these general categories of maps, there are maps with special properties; these may or may not form categories, and may or may not be generally discussed categorically.

[[File:TrefoilKnot 01.svg|thumb|The right-handed [[trefoil knot]].]]
In [[geometric topology]] a basic type are [[embedding]]s, of which [[knot theory]] is a central example, and generalizations such as [[immersion (mathematics)|immersion]]s, [[Submersion (mathematics)|submersions]], [[covering space]]s, and [[ramified covering space]]s.
Basic results include the [[Whitney embedding theorem]] and [[Whitney immersion theorem]].

[[Image:Riemann_sqrt.jpg|thumb|right|[[Riemann surface]] for the function ''&fnof;''(''z'') = √''z'', shown as a [[ramified covering space]] of the complex plane.]]
In complex geometry, ramified covering spaces are used to model [[Riemann surface]]s, and to analyze maps between surfaces, such as by the [[Riemann–Hurwitz formula]].

In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of [[isometric embedding]]s, [[isometric immersion]]s, and [[Riemannian submersion]]s; a basic result is the [[Nash embedding theorem]].

== Scalar-valued functions ==
[[Image:Spherical harmonics.png|300px|thumb|right|3D color plot of the [[spherical harmonics]] of degree <math>n=5</math>]]
A basic example of maps between manifolds are scalar-valued functions on a manifold, <math>\scriptstyle f\colon M \to \mathbb{R}</math> or <math>\scriptstyle f\colon M \to \mathbb{C},</math> sometimes called [[regular function]]s or [[functional (mathematics)|functional]]s, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.

In geometric topology, most commonly studied are [[Morse function]]s, which yield [[handlebody]] decompositions, which generalize to [[Morse–Bott function]]s and can be used for instance to understand classical groups, such as in [[Bott periodicity]].

In [[mathematical analysis]], one often studies solution to [[partial differential equations]], an important example of which is [[harmonic analysis]], where one studies [[harmonic function]]s: the kernel of the [[Laplace operator]]. This leads to such functions as the [[spherical harmonics]], and to [[heat kernel]] methods of studying manifolds, such as [[hearing the shape of a drum]] and some proofs of the [[Atiyah–Singer index theorem]].

The [[monodromy]] around a [[mathematical singularity|singularity]] or [[branch point]] is an important part of analyzing such functions.

== Curves and paths ==
[[Image:Football3c.jpg|right|thumb|200px|A [[geodesic]] on an [[Football (ball)|American football]] illustrating the proof of Gromov's [[filling area conjecture]] in [[systolic geometry]], in the hyperelliptic case (see [[Systolic geometry#Filling area conjecture|explanation]]).]]
Dual to scalar-valued functions – maps <math>\scriptstyle M \to \mathbb{R}</math> – are maps <math>\scriptstyle \mathbb{R} \to M,</math> which correspond to curves or paths in a manifold. One can also define these where the domain is an interval <math>[a,b],</math> especially the [[unit interval]] <math>[0,1],</math> or where the domain is a circle (equivalently, a periodic path) ''S''<sup>1</sup>, which yields a loop. These are used to define the [[fundamental group]], [[Chain (algebraic topology)|chains]] in [[homology theory]], [[geodesic]] curves, and [[systolic geometry]].

Embedded paths and loops lead to [[knot theory]], and related structures such as [[Link (knot theory)|links]], [[Braid theory|braids]], and [[Tangle (mathematics)|tangles]].

== Metric spaces ==
Riemannian manifolds are special cases of [[metric spaces]], and thus one has a notion of [[Lipschitz continuity]], [[Hölder condition]], together with a [[coarse structure]], which leads to notions such as coarse maps and connections with [[geometric group theory]].

== See also ==
* [[:Category:Maps of manifolds]]

[[Category:Maps of manifolds| ]]

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en>Braddarb at 00:10, 24 January 2014