A pseudo-Riemannian manifold is a manifold $M$ together with a non degenerate, 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 result from linear algebra 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)¹.

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.

Footnotes

... (1,3)¹

also referred to as $(-+++)$