Definition:Pseudo-Riemannian Metric

Definition
Let $M$ be a smooth manifold.

Let $p \in M$ be a point in $M$.

Let $T_p M$ be the tangent space of $M$ at $p$ with the scalar product $\innerprod \cdot \cdot_p$.

Let $g \in \map {\TT^2} M$ be a smooth symmetric 2-tensor field such that for all $p$ its value at $p$ is equal to $\innerprod \cdot \cdot_p$:


 * $\forall p \in M : g_p = \innerprod \cdot \cdot_p$

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
The pseudo-Riemannian metric is also known as semi-Riemannian metric.