\begin{definition}[Definition:Pseudo-Riemannian Metric]
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'''.
\end{definition}