Flat metric - Revision history

Vipul: 8 revisions

2008-05-18T19:40:21Z

8 revisions

← Older revision	Revision as of 19:40, 18 May 2008
(No difference)

Vipul at 09:39, 2 September 2007

2007-09-02T09:39:53Z

← Older revision		Revision as of 09:39, 2 September 2007
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	* [[Einstein metric]]		* [[Einstein metric]]
	* [[Constant-scalar curvature metric]]		* [[Constant-scalar curvature metric]]
			* [[Conformally flat metric]]

	==Facts==		==Facts==

Vipul: /* Holonomy */

2007-09-02T03:21:20Z

Holonomy

← Older revision		Revision as of 03:21, 2 September 2007
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	===Holonomy===		===Holonomy===

	The [[restricted holonomy group of Riemannian metric]] of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial [[holonomy of a loop\|holonomy]]. Thus, the [[holonomy group]] is a homomorphic image of the [[fundamental group]], and gives rise to a [[linear representation]] of the fundamental group.		The [[restricted holonomy group of Riemannian metric\|restricted holonomy group]] of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial [[holonomy of a loop\|holonomy]]. Thus, the [[holonomy group]] is a homomorphic image of the [[fundamental group]], and gives rise to a [[linear representation]] of the fundamental group.

	===Minding's theorem===		===Minding's theorem===

Vipul: /* Holonomy */

2007-09-02T03:20:58Z

Holonomy

← Older revision		Revision as of 03:20, 2 September 2007
Line 25:		Line 25:
	===Holonomy===		===Holonomy===

	The [[[restricted holonomy group of Riemannian metric]] of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial [[holonomy of a loop\|holonomy]]. Thus, the [[holonomy group]] is a homomorphic image of the [[fundamental group]], and gives rise to a [[linear representation]] of the fundamental group.		The [[restricted holonomy group of Riemannian metric]] of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial [[holonomy of a loop\|holonomy]]. Thus, the [[holonomy group]] is a homomorphic image of the [[fundamental group]], and gives rise to a [[linear representation]] of the fundamental group.

	===Minding's theorem===		===Minding's theorem===

Vipul: /* =Facts */

2007-09-02T03:03:12Z

=Facts

← Older revision		Revision as of 03:03, 2 September 2007
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	* [[Constant-scalar curvature metric]]		* [[Constant-scalar curvature metric]]

	==Facts=		==Facts==

	===Holonomy===		===Holonomy===

Vipul: /* Relation with other properties */

2007-09-02T03:00:58Z

Relation with other properties

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 * [[Einstein metric]]
 * [[Constant-scalar curvature metric]]
 ==Metaproperties==

Vipul: /* Symbol-free definition */

2007-09-02T02:47:21Z

Symbol-free definition

← Older revision		Revision as of 02:47, 2 September 2007
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	===Symbol-free definition===		===Symbol-free definition===

	A [[Riemannian metric]] on a [[differential manifold]] is said to be a '''flat metric''' if the [[sectional curvature]] is identically zero, viz the sectional curvature is zero for every tangent plane at every point.		A [[Riemannian metric]] on a [[differential manifold]] is said to be a '''flat metric''' if it satisfies the following equivalent conditions:

			* The [[sectional curvature]] is identically zero, viz the sectional curvature is zero for every tangent plane at every point.
			* The [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor]] of the [[Levi-Civita connection]] is the zero map, viz vanishes everywhere

	==Relation with other properties==		==Relation with other properties==

Vipul at 07:56, 25 April 2007

2007-04-25T07:56:43Z

← Older revision		Revision as of 07:56, 25 April 2007
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	==Metaproperties==		==Metaproperties==

	{{S-closed}}		{{DP-closed metric property}}

	~~Given a flat~~ metric ~~on a differential manifold~~
	~~{{DP-closed~~}}

	The direct product of two Riemannian manifolds, both equipped with a flat metric, also gets the flat metric. Thus, since every curve is a flat manifold, any manifold obtained as a direct product of curves is also flat.		The direct product of two Riemannian manifolds, both equipped with a flat metric, also gets the flat metric. Thus, since every curve is a flat manifold, any manifold obtained as a direct product of curves is also flat.

Vipul at 07:52, 25 April 2007

2007-04-25T07:52:30Z

New page

{{Riemannian metric property}}

{{zeroproperty|[[sectional curvature]]}}

==Definition==

===Symbol-free definition===

A [[Riemannian metric]] on a [[differential manifold]] is said to be a '''flat metric''' if the [[sectional curvature]] is identically zero, viz the sectional curvature is zero for every tangent plane at every point.

==Relation with other properties==

===Weaker properties===

* [[Constant-curvature metric]]
* [[Ricci-flat metric]]
* [[Einstein metric]]
* [[Constant-scalar curvature metric]]

==Metaproperties==

{{S-closed}}

Given a flat metric on a differential manifold
{{DP-closed}}

The direct product of two Riemannian manifolds, both equipped with a flat metric, also gets the flat metric. Thus, since every curve is a flat manifold, any manifold obtained as a direct product of curves is also flat.