# pseudo-Riemannian manifold

A pseudo-Riemannian manifold is a manifold $M$ together with a non degenerate (http://planetmath.org/NonDegenerateBilinearForm), symmetric section $g$ of $T^{0}_{2}(M)$ (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 $g(v,v)\leq 0$.

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)11also referred to as $(-+++)$.

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