# 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)^{1}^{1}also 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 |