Flat metric: Difference between revisions
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* [[Einstein metric]] | * [[Einstein metric]] | ||
* [[Constant-scalar curvature metric]] | * [[Constant-scalar curvature metric]] | ||
==Facts= | |||
===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. | |||
===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== | ||
Revision as of 03:00, 2 September 2007
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
=Facts
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. 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.