PSEUDO-RIEMANNIAN MANIFOLD

In differential geometry, a 'pseudo-Riemannian manifold' (also called a 'semi-Riemannian manifold') is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann. The key difference between the two is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed.
Arguably, the most important type of pseudo-Riemannian manifold is a 'Lorentzian manifold'. Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime. Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space.

Contents

Formal definition

Lorentzian manifolds

Properties of pseudo-Riemannian manifolds

Formal definition

A 'pseudo-Riemannian manifold' is a smooth manifold equipped with a smooth, symmetric $(0,2)$ tensor which is nondegenerate at each point on the manifold. This tensor is called a 'pseudo-Riemannian metric' or, simply, a '(pseudo-)metric tensor'.
Every nondegenerate, symmetric, bilinear form on a
vector space can be assigned a signature $(p,q)$. Here $p$ and $q$ denote the number of positive and negative eigenvalues of the form. The 'signature' of a pseudo-Riemannian manifold is just the signature of the metric on any given tangent space (one should insist that the signature is the same on every connected component). Note that $p\; +\; q\; =\; n$ is the dimension of the manifold. A Riemannian metric has signature $(n,0)$.

Lorentzian manifolds

Pseudo-Riemannian metrics of signature $(p,1)$ (or sometimes $(1,q)$, see sign convention) are called 'Lorentzian metrics'. A manifold equipped with a Lorentzian metric is naturally called a 'Lorentzian manifold'. After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of general relativity. A principal assumption of general relativity is that spacetime can be modeled as a Lorentzian manifold of signature $(3,1)$.

Properties of pseudo-Riemannian manifolds

Just as Euclidean space 'R'^''n'' can be thought of as the model Riemannian manifold, Minkowski space 'R'^''p'',1 with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature $(p,q)$ is 'R'^''p'',''q'' with the metric
:$g\; =\; dx\_1^2\; +\; cdots\; +\; dx\_p^2\; -\; dx\_\{p+1\}^2\; -\; cdots\; -\; dx\_\{p+q\}^2$
Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well. This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor. On the other hand, there are many theorems in Riemannian geometry which do not hold in the generalized case. For example, it is ''not'' true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain topological obstructions.