pseudo-Riemannian manifold

A pseudo-Riemannian manifold is a manifold $M$ together with a non degenerate (http://planetmath.org/NonDegenerateBilinearForm), symmetric section $g$ of $T_2^0(M)$ (2-covariant tensor bundle over $M$).

Unlike with a Riemannian manifold, $g$ is not positive definite. That is, there exist vectors $v\in T_{p}M$ such that $g(v,v)\le 0$.

A well known result from linear algebra (http://planetmath.org/SylvestersLaw) permits us to make a change of basis such that in the new base $g$ is represented by a diagonal matrix with $-1$ or $1$ elements in the diagonal. If there are $i$, $-1$ elements in the diagonal and $j$, $1$, the tensor is said to have signature $(i,j)$

The signature will be invariant in every connected component of $M$, but usually the restriction that it be a global invariant is added to the definition of a pseudo-Riemannian manifold.

Unlike a Riemannian metric, some manifolds do not admit a pseudo-Riemannian metric.

Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold with signature (1,3).

Intuitively pseudo-Riemannian manifolds are generalizations of Minkowski’s space just as a Riemannian manifold is a generalization of a vector space with a positive definite metric.

Title	pseudo-Riemannian manifold
Canonical name	PseudoRiemannianManifold
Date of creation	2013-03-22 15:44:15
Last modified on	2013-03-22 15:44:15
Owner	cvalente (11260)
Last modified by	cvalente (11260)
Numerical id	10
Author	cvalente (11260)
Entry type	Definition
Classification	msc 53Z05
Related topic	EinsteinFieldEquations
Related topic	SylvestersLaw
Related topic	MinkowskiSpace
Related topic	CategoryOfRiemannianManifolds
Defines	pseudo-Riemannian geometry
Defines	pseudo-Riemannian manifold