An infinitely differentiable function that is not analytic (3215 bytes)
9: ...f''^(''n'') we prove to exist. For any differentiable function ''R''(''x'') the derivative for ''x'' > ...
34: ...rentiable function one can get another infinitely differentiable function with prescribed behavior <!-- prescribed...
36: ...of differentiable functions on a [[differentiable manifold]] is [[flasque]], in contrast with the analytic c...
38: ... of unity|partitions of unity]] on differentiable manifolds.

4-manifold (2183 bytes)
1: ...'''4-manifold''' is a 4 dimensional topological [[manifold]].
3: A closed 4-manifold M is usually described by a [[handle decompositio...
9: ...us]]; since this solid torus is embedded in a [[3-manifold]], there
10: ...ositions on 4-manifolds, and [[knot theory]] in 3-manifolds.
13: ... can be cancelled without changing the underlying manifold. Similarly, such

G2 manifold (998 bytes)
1: {{wrongtitle|title=G₂ manifold}}
3: ...ensional spinor representation. ''G''₂ manifolds are [[Ricci-flat]]. The name is for [[Dominic Jo...
5: ..., [[M-theory]] compactified on a <math>G_2</math> manifold leads to a realistic four-dimensional (11-7=4) th...
7: ''See also'': [[Calabi-Yau manifold]], [[Spin(7) manifold]]

Manifold (15505 bytes)
3: In [[mathematics]], a '''manifold''' ''M'' is a type of [[space]], characterised in...
8: ...e the surface of the Earth is a (two-dimensional) manifold.
12: ...hen considered geometrically, were also naturally manifold theories, with a concept of [[generalized coordin...
16: ...It is also possible and very useful to describe a manifold from the point of view of an outside observer. Fo...
18: ...e wrapped in different ways round [[cylinder]]s). Manifolds include familiar [[curve]]s such as the [[circle...

3-manifold (3805 bytes)
1: ...ng with say, topological 3-manifolds, or smooth 3-manifolds.
3: ...heory|TQFTs]], and [[gauge theory]]. Sometimes 3-manifold theory is referred to as [[low-dimensional topolo...
5: ...ncompressible surface]] and the theory of [[Haken manifold]]s, or one can choose the complementary pieces to...
9: ...tric and topological information belonging to a 3-manifold. Thus, there is an interplay between [[group the...
11: ==Some famous examples of 3-manifolds==

Hyperkähler manifold (1348 bytes)
1: ...]] analogues of Kähler manifolds. All hyperkähler manifolds are [[Ricci-flat]].
3: Every hyperkähler manifold ''M'' has a [[sphere|2-sphere]] of [[complex stru...
21: ... [[algebraic geometry]]. Noncompact hyperkähler 4-manifolds which are asymptotic to '''H'''/''G'', where '''...
23: ''See also'': [[Calabi-Yau manifold]]

Sub-Riemannian manifold (2102 bytes)
1: ...peaking, to measure distances in a sub-Riemannian manifold,
4: ...l dimension]] (unless it is actually a Riemannian manifold).
18: A '''sub-Riemannian manifold''' is a triple <math>(M, H, g)</math>,
19: where <math>M,</math> is a differentiable [[manifold]],
23: Any '''sub-Riemannian manifold''' carries the natural [[intrinsic metric]], call...

Almost flat manifold (1044 bytes)
1: In [[mathematics]], a smooth [[compact]] [[manifold]] ''M'' is called '''almost flat''' if for any <m...
4: ...ilon_n>0 </math> such that if a ''n''-dimensional manifold admits an <math>\epsilon_n</math>-flat metric wit...
6: ...Glossary_of_Riemannian_and_metric_geometry#N|nill manifold]], i.e. a total space of a oriented circle [[bund...

Poisson manifold (768 bytes)
1: A '''Poisson manifold''' is a [[differential manifold]] M such that the algebra of [[smooth function]]s...
3: Every [[symplectic manifold]] is a Poisson manifold but not vice versa.
5: ...eta;(dh,df),dg) for all f, g, h. For a symplectic manifold, &eta; is nothing other than the inverse of the s...
7: ...e also [[Poisson supermanifold]], [[Nambu-Poisson manifold]]

Stein manifold (1443 bytes)
1: ...sed set|closed]], [[complex number|complex]] [[submanifold]] of the vector space of ''n'' complex dimensions...
3: ...ion 1, a connected [[Riemann surface]] is a Stein manifold if and only if it is not compact.
5: Numerous further characterizations of such manifolds exist, in particular capturing the property of t...
7: In the [[GAGA]] set of analogies, Stein manifolds correspond to [[affine variety|affine varieties]...
9: ...umbers into themselves. It is known that a Stein manifold is elliptic if and only if it is [[fibrant object...

Hyperbolic 3-manifold (2021 bytes)
1: A '''hyperbolic 3-manifold''' is a [[3-manifold]] equipped with a [[complete]] [[Riemannnian metr...
3: ...yperbolic knots result in [[closed]] hyperbolic 3-manifolds.
5: ...celebrated [[geometrization theorem]] for [[Haken manifolds]].
7: ...erites.) Heuristically, this means that "many" 3-manifolds are in fact hyperbolic.
11: [[Category:3-manifolds]][[Category:Riemannian geometry]]

Manifold decomposition (3352 bytes)
1: ...f [[mathematics]], the '''decomposition''' of a [[manifold]] ''M'' is writing ''M'' as a combination of smal...
3: ...r, it has been useful in attempts to classify [[3-manifold]]s and also in proving the higher dimensional [[P...
5: ...the smaller pieces are combined to make the large manifold.
16: ...p to common subdivision. In dimension 4, not all manifolds are triangulable. For higher dimensions, genera...
23: ...]], [[orientable]], [[Compact space|compact]] [[3-manifold]]s

Spherical 3-manifold (1112 bytes)
1: ...entable]], [[Manifold|closed]] [[3-manifold|3]]-[[manifold]] of the form
5: ...al 3-manifolds are sometimes called '''elliptic 3-manifolds'''.
7: ... finite fundamental group, then it is a spherical manifold.
9: ...by its fundamental group, but any other spherical manifold is.
11: [[Category:Riemannian geometry]][[Category:3-manifolds]]

Whitehead manifold (2732 bytes)
1: ...cs]], the '''Whitehead manifold''' is an open [[3-manifold]] that is [[contractible]], but not [[homeomorphi...
3: ...l are contractible, too. Are ''all'' contractible manifolds homeomorphic to a ball? For dimensions 1 and 2, ...
5: ...ents the first [[counterexample]]:- the Whitehead manifold.
15: ...g space is S³\W which is a non-compact manifold without boundary. <math>W \times \mathbb R \cong...
17: More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fash...

Manifold (disambiguation) (403 bytes)
1: The word '''manifold''' has a variety of meanings.
3: *In [[mathematics]], a [[manifold]] is a [[topological space]] that locally "looks ...
4: ...ineering]], a [[manifold (automotive engineering)|manifold]] is an [[engine]] part.
5: *The [[River Manifold]] is a [[river]] in [[England]]
6: *[[John Manifold]] was an [[Australia]]n [[poet]] and critic.

John Manifold (1690 bytes)
1: '''John Streeter Manifold''' ([[21 April]] [[1915]] – [[19 April]] [[1985]]...
20: [[Category:Australian poets|Manifold, John]]

Einstein manifold (800 bytes)
1: An '''Einstein manifold''' is a [[Riemannian manifold]] whose [[Ricci tensor]] is proportional to the [...
3: ...ty]], these manifolds (in the [[pseudo-Riemannian manifold|pseudo-Riemannian]] case) can be thought of as [[...
5: ...folds with ''k'' = 0 are also called [[Ricci-flat manifold]]s.

Kähler manifold (2612 bytes)
1: ...t of as [[Riemannian manifold]]s and [[symplectic manifold]]s in a natural way.
3: Kähler manifolds are named for the mathematician [[Erich Kähler]]...
7: A '''Kähler metric''' on a complex manifold ''M'' is a [[hermitian metric]] on the [[complexi...
11: ... If M carries such a metric it is called a Kähler manifold.
15: ...p> with the standard Hermitian metric is a Kähler manifold.

Flag manifold (4506 bytes)
17: a '''flag manifold'''.
21: ...al or complex field we can also consider the flag manifolds as coset spaces for [[Lie group]]s - in more tha...
27: ...mensions, are then referred to as '''partial flag manifolds'''.
29: ==Stiefel manifolds==
31: ...ace. There is an important mapping from a Stiefel manifold to the Grassmannian forgetting the extra informat...

Haken manifold (3296 bytes)
1: ... manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surfa...
3: ...of incompressible surfaces. He proved that Haken manifolds have a '''hierarchy'''.
5: ...So the regular neighborhood is a 3-dimensional submanifold with boundary being two copies of the surface.
7: ... eventually this sequence of cutting results in a manifold whose pieces (or components) are just 3-balls, we...
9: Summing up, Haken proved that given any Haken manifold, there is a (finite) sequence of cuttings (as abo...

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