Lorentzian manifold: Difference between revisions
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A '''Lorentzian manifold''' is the following data: | A '''Lorentzian manifold''' is the following data: | ||
* A [[differential manifold]] <math | * A [[differential manifold]] <math>M</math> | ||
* A symmetric nondegenerate bilinear form of type <math>(n,1)</math> on each tangent space, such that the form varies smoothly with the point. | * A symmetric nondegenerate bilinear form of type <math>(n,1)</math> on each tangent space, such that the form varies smoothly with the point. | ||
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Lorentzian manifolds are special cases of [[pseudo-Riemannian manifold]]s. Thus, all the generic constructs for a pseudo-Riemannian manifold apply to Lorentzian manifolds. | Lorentzian manifolds are special cases of [[pseudo-Riemannian manifold]]s. Thus, all the generic constructs for a pseudo-Riemannian manifold apply to Lorentzian manifolds. | ||
==Relation with other structures== | |||
===Weaker structures=== | |||
* [[Pseudo-Riemannian manifold]] | |||
Latest revision as of 19:48, 18 May 2008
This article defines a differential manifold with the following additional structure -- the structure group is reduced to: Lorentzian group viz a group of the form
Definition
A Lorentzian manifold is the following data:
- A differential manifold
- A symmetric nondegenerate bilinear form of type on each tangent space, such that the form varies smoothly with the point.
In other words, a Lorentzian manifold is the reduction of the structure group of a differential manifold to the Lorentzian group .
Lorentzian manifolds are special cases of pseudo-Riemannian manifolds. Thus, all the generic constructs for a pseudo-Riemannian manifold apply to Lorentzian manifolds.