Flat metric: Difference between revisions
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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== | ||
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* [[Einstein metric]] | * [[Einstein metric]] | ||
* [[Constant-scalar curvature metric]] | * [[Constant-scalar curvature metric]] | ||
* [[Conformally flat metric]] | |||
==Facts== | |||
===Holonomy=== | |||
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=== | |||
{{further|[[Minding's theorem]]}} | |||
This result says that two smooth surfaces with the same constant curvature are [[locally isometric Riemannian manifolds|locally isometric]]. | |||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 19:40, 18 May 2008
This article defines a property that makes sense for a Riemannian metric over a differential manifold
This is the property of the following curvature being everywhere zero: sectional curvature
Definition
Symbol-free definition
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 the Levi-Civita connection is the zero map, viz vanishes everywhere
Relation with other properties
Weaker properties
- Constant-curvature metric
- Ricci-flat metric
- Einstein metric
- Constant-scalar curvature metric
- Conformally flat metric
Facts
Holonomy
The restricted holonomy group of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial 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
Further information: Minding's theorem
This result says that two smooth surfaces with the same constant curvature are locally isometric.
Metaproperties
Direct product-closedness
This property of a Riemannian metric on a differential manifold is closed under taking direct products of manifolds
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.