pseudo-Riemannian manifold PseudoRiemannianManifold 2006-03-06 16:08:31 2007-08-07 18:47:17 Definition pseudo-Riemannian geometry pseudo-Riemannian manifold % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{pseudo-Riemannian} manifold is a manifold $M$ together with a \PMlinkname{non degenerate}{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 $g(v,v)\le0$. A well known \PMlinkname{result from linear algebra}{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)\footnote{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.