Constant-curvature metric: Difference between revisions
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{{Riemannian metric property}} | {{Riemannian metric property}} | ||
{{constancyproperty|[[sectional curvature]]}} | |||
==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[Riemannian metric]] on a [[ | A [[Riemannian metric]] on a [[differential manifold]] is termed a '''constant-curvature metric''' if, for any section of the manifold, the sectional curvature is constant at all points, and moreover, this constant value is the same for all sections. | ||
Equivalently, a Riemannian metric on a differential manifold is termed a constant-curvature metric if it satisfies the following, termed the '''axiom of free mobility''', namely: given any two points in the manifold, and any orthonormal bases for the tangent spaces at the two points, there are neighbourhoods of the two points and a Riemannian isometry from one to the other, that maps one orthonormal basis to the other. | |||
==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | |||
* [[Flat metric]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
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* [[Einstein metric]] | * [[Einstein metric]] | ||
* [[Locally homogeneous metric]] | * [[Locally homogeneous metric]] | ||
* [[Constant-scalar curvature metric]] | |||
For manifolds of dimension upto three, constant-curvature metrics are the same as Einstein metrics. | For manifolds of dimension upto three, constant-curvature metrics are the same as Einstein metrics. | ||
Latest revision as of 19:36, 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 constant: sectional curvature
Definition
Symbol-free definition
A Riemannian metric on a differential manifold is termed a constant-curvature metric if, for any section of the manifold, the sectional curvature is constant at all points, and moreover, this constant value is the same for all sections.
Equivalently, a Riemannian metric on a differential manifold is termed a constant-curvature metric if it satisfies the following, termed the axiom of free mobility, namely: given any two points in the manifold, and any orthonormal bases for the tangent spaces at the two points, there are neighbourhoods of the two points and a Riemannian isometry from one to the other, that maps one orthonormal basis to the other.
Relation with other properties
Stronger properties
Weaker properties
For manifolds of dimension upto three, constant-curvature metrics are the same as Einstein metrics.