What is "pseudo-riemannian manifold"

n. in differential geometry, a generalization of a Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but is instead only required to be nondegenerate, which is a weaker condition.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean space described by a quadratic form, which may be isotropic.

A special case of great importance to general relativity is a Lorentzian manifold, in which one dimension has a sign opposite to that of the rest. This allows tangent vectors to be classified into timelike, null, and spacelike. Spacetime can be modeled as a 4-dimensional Lorentzian manifold.