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== | ||
Revision as of 02:47, 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
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.