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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 (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 (one should insist that the signature is the same on every connected component). Note that p + q = n is the dimension of the manifold. Riemannian manifolds are simply those with signature (n,0).

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 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).

Just as Euclidean space Rⁿ can be thought of as the model Riemannian manifold, Minkowski spaceIn physics and mathematics, Minkowski space (or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a si 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

Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the fundamental theorem of Riemannian geometryIn Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor. Such a connection is called a Levi-Civit is true of pseudo-Riemannian manifolds has well. This allows one to speak of the Levi-Civita connectionIn Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). The Fundamental theorem of Riemannian geometry stat on a pseudo-Riemannian manifold along with the associated curvature tensorIn differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds. The curvature tensor is given in terms of a Levi-Civita connection (or covariant differentiation) by the following formula: : Her. 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 topologicalTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g obstructions.