Space of Riemannian metrics - Revision history

Vipul: 6 revisions

2008-05-18T20:09:47Z

6 revisions

← Older revision	Revision as of 20:09, 18 May 2008
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Vipul: /* Riemannian metrics upto diffeomorphism equivalence */

2007-07-06T15:26:10Z

Riemannian metrics upto diffeomorphism equivalence

← Older revision		Revision as of 15:26, 6 July 2007
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	===Riemannian metrics upto diffeomorphism equivalence===		===Riemannian metrics upto diffeomorphism equivalence===

	We declare two Riemannian metrics to be diffeomorphically equivalent if there is a diffeomorphism of the underlying manifold that takes one Riemannian metric to the other. The moduli space obtaiend by quotienting out by this equivalence relation is termed the ~~'''~~space of Riemannian metrics up to diffeomorphism~~'''~~.		We declare two Riemannian metrics to be diffeomorphically equivalent if there is a diffeomorphism of the underlying manifold that takes one Riemannian metric to the other. Here, we use the fact that when there is a local diffeomorphism, we can pull back Riemannian metrics in a unique way. The moduli space obtaiend by quotienting out by this equivalence relation is termed the [[space of Riemannian metrics up to diffeomorphism]].

	==Points in the space of Riemannian metrics==		==Points in the space of Riemannian metrics==

Vipul: /* Subspaces obtained by curvature bounds */

2007-07-06T15:25:02Z

Subspaces obtained by curvature bounds

← Older revision		Revision as of 15:25, 6 July 2007
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	===Subspaces obtained by curvature bounds===		===Subspaces obtained by curvature bounds===

	Given two (extended) real numbers <math>a</math> and <math>b</math>, we can consider the subset of the space of Riemannian metrics comprising those Riemannian metrics for which the sectional curvature everywhere lies in <math>[a,b]</math> or <math>[a,b)</math> or <math>(a,b]</math> or <math>(a,b)</math>. Some particular examples are:		Given two (extended) real numbers <math>a</math> and <math>b</math>, we can consider the subset of the space of Riemannian metrics comprising those Riemannian metrics for which the [[sectional curvature]] everywhere lies in <math>[a,b]</math> or <math>[a,b)</math> or <math>(a,b]</math> or <math>(a,b)</math>. Some particular examples are:

	* <math>(-\infty,0)</math> describes the [[space of negatively curved metrics]]		* <math>(-\infty,0)</math> describes the [[space of negatively curved metrics]]
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	* <math>[-1,-1]</math> describes the [[space of hyperbolic metrics]]		* <math>[-1,-1]</math> describes the [[space of hyperbolic metrics]]

			The [[Ricci flow on surfaces]] actually shows that for a surface, the space of hyperbolic metrics is a deformation retract of the space of negatively curved metrics.

	==Quotients and moduli==		==Quotients and moduli==

Vipul: /* Quotients and moduli */

2007-07-06T15:21:53Z

Quotients and moduli

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 The space of Riemannian metrics is important because we are interested in studying evolutions of Riemannian metrics under [[flow equation]]s, and the manifold over which these flow equations are considered is precisely the space of Riemannian metrics.
 ==Quotients and moduli==
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 ===Riemannian metrics upto conformal equivalence===
-We declare two Riemannian metrics to be conformally equivalent if there is a scalar, everywhere nonzero function that multiplied with one Riemannian metric, gives the other. This is essentially equivalence under the action of the multiplicative group of everywhere positive functions, on the tensor bundle. The quotient of the space of Riemannian metrics under the equivalence relation of conformal equivalence, is termed the '''space of conformal classes of Riemannian metrics'''.
+We declare two Riemannian metrics to be conformally equivalent if there is a scalar, everywhere nonzero function that multiplied with one Riemannian metric, gives the other. This is essentially equivalence under the action of the multiplicative group of everywhere positive functions, on the tensor bundle. The quotient of the space of Riemannian metrics under the equivalence relation of conformal equivalence, is termed the [[space of conformal classes of Riemannian metrics]].
 ===Riemannian metrics upto diffeomorphism equivalence===

Vipul: /* Definition part */

2007-07-06T15:17:06Z

Definition part

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	===Definition part===		===Definition part===

	The '''space of Riemannian metrics''' on <math>M</math> is the set of all Riemannian metrics on <math>M</math>. Since each Riemannian metric can be viewed as a <math>(0,2)</math>-tensor (viz a section of the <math>(0,2)</math>-tensor bundle), the space of Riemannian metrics can be viewed as a subset of the space of all sections of the <math>(0,2)</math>-tensor bundle.		The '''space of Riemannian metrics''' on <math>M</math> is the set of all Riemannian metrics on <math>M</math>. Since each Riemannian metric can be viewed as a <math>(0,2)</math>-tensor (viz a section of the <math>(0,2)</math>-tensor bundle), the space of Riemannian metrics can be viewed as a subset of the space of all sections of the <math>(0,2)</math>-tensor bundle.

			The space of Riemannian metrics is studied as a topological space, with the smooth topology.

	The space of Riemannian metrics is important because we are interested in studying evolutions of Riemannian metrics under [[flow equation]]s, and the manifold over which these flow equations are considered is precisely the space of Riemannian metrics.		The space of Riemannian metrics is important because we are interested in studying evolutions of Riemannian metrics under [[flow equation]]s, and the manifold over which these flow equations are considered is precisely the space of Riemannian metrics.

Vipul at 13:29, 8 April 2007

2007-04-08T13:29:24Z

← Older revision		Revision as of 13:29, 8 April 2007
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	We declare two Riemannian metrics to be diffeomorphically equivalent if there is a diffeomorphism of the underlying manifold that takes one Riemannian metric to the other. The moduli space obtaiend by quotienting out by this equivalence relation is termed the '''space of Riemannian metrics up to diffeomorphism'''.		We declare two Riemannian metrics to be diffeomorphically equivalent if there is a diffeomorphism of the underlying manifold that takes one Riemannian metric to the other. The moduli space obtaiend by quotienting out by this equivalence relation is termed the '''space of Riemannian metrics up to diffeomorphism'''.

			==Points in the space of Riemannian metrics==

			The question: ''given a [[differential manifold]], can it be equipped with a metric having a given property?'' translates to the question: ''given a [[differential manifold]], is there a point in its space of Riemannian metrics that satisfies the given property?''

			To prove that there does exist a point satisfying the given property, one of the tools we use is to consider a [[differential operator]] on the space of Riemannian metrics such that the Riemannian metrics it maps to zero are precisely those having the property. We then look at the [[flow equation]] for this differential operator, and try to find a trajectory for the flow equation that converges at <math>\infty</math> or <math>-\infty</math>.

			{{further\|[[Flow method to prove existence of certain metrics]]}}

Vipul at 13:12, 8 April 2007

2007-04-08T13:12:15Z

New page

==Definition==

===Given data===

A [[differential manifold]] <math>M</math>.

===Definition part===

The '''space of Riemannian metrics''' on <math>M</math> is the set of all Riemannian metrics on <math>M</math>. Since each Riemannian metric can be viewed as a <math>(0,2)</math>-tensor (viz a section of the <math>(0,2)</math>-tensor bundle), the space of Riemannian metrics can be viewed as a subset of the space of all sections of the <math>(0,2)</math>-tensor bundle.

The space of Riemannian metrics is important because we are interested in studying evolutions of Riemannian metrics under [[flow equation]]s, and the manifold over which these flow equations are considered is precisely the space of Riemannian metrics.

==Quotients and moduli==

===Riemannian metrics upto conformal equivalence===

We declare two Riemannian metrics to be conformally equivalent if there is a scalar, everywhere nonzero function that multiplied with one Riemannian metric, gives the other. This is essentially equivalence under the action of the multiplicative group of everywhere positive functions, on the tensor bundle. The quotient of the space of Riemannian metrics under the equivalence relation of conformal equivalence, is termed the '''space of conformal classes of Riemannian metrics'''.

===Riemannian metrics upto diffeomorphism equivalence===

We declare two Riemannian metrics to be diffeomorphically equivalent if there is a diffeomorphism of the underlying manifold that takes one Riemannian metric to the other. The moduli space obtaiend by quotienting out by this equivalence relation is termed the '''space of Riemannian metrics up to diffeomorphism'''.