Pseudo-Riemannian manifold

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In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, <math>(0,2)<math> tensor which is nondegenerate at each point on the manifold. This tensor is called a pseudo-Riemannian metric or, simply, a (pseudo-)metric tensor.

The key difference between a Riemannian metric and a pseudo-Riemannian metric is that a pseudo-Riemannian metric need not be positive-definite, merely nondegenerate. Since every positive-definite form is also nondegenerate a Riemannian metric is a special case of a pseudo-Riemannian one. Thus pseudo-Riemannian manifolds can be considered generalizations of Riemannian manifolds.

Every nondegenerate, symmetric, bilinear form has a fixed signature <math>(p,q)<math>. Here <math>p<math> and <math>q<math> 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 (one should insist that the signature is the same on every connected component). Note that <math>p + q = n<math> is the dimension of the manifold. Riemannian manifolds are simply those with signature <math>(n,0)<math>.

Pseudo-Riemannian metrics of signature <math>(p,1)<math> (or sometimes <math>(1,q)<math>, 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 <math>(3,1)<math>.

Just as Euclidean space <math>\mathbf R^n<math> can be thought of as the model Riemannian manifold, Minkowski space <math>\mathbf R^{p,1}<math> with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature <math>(p,q)<math> is <math>\mathbf R^{p,q}<math> with the metric

<math>g = dx_1^2 + \cdots + dx_p^2 - dx_{p+1}^2 - \cdots - dx_{p+q}^2<math>

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.


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