Definition:Pseudo-Riemannian Metric

Definition
Let $M$ be a smooth manifold.

Let $g$ be a smooth symmetric 2-tensor field on $M$.

Suppose $g$ is nondegenerate.

Suppose for all $p \in M$ the signature of $g_p$ is the same.

Then $g$ is called the pseudo-Riemannian metric.

Also known as
Pseudo-Riemannian metric is also known as semi-Riemannian metric.