\begin{definition}[Definition:Pseudo-Euclidean Space of Signature (r,s)]
Let $r, s \in \N$ be natural numbers.
Let $\R^{r + s}$ be an $\paren {r + s}$-dimensional real vector space.
Let $\tuple {\xi^1 \ldots \xi^r, \tau^1 \ldots \tau^s}$ be standard coordinates of $\R^{r + s}$.
Let $\bar q^{\paren {r, s}}$ be a pseudo-Riemannian metric defined by:
:$\bar q^{\paren {r, s}} = \paren {\d \xi^1}^2 + \ldots \paren {\d \xi^r}^2 - \paren {\d \tau^1}^2 - \ldots - \paren {\d \tau^s}^2$
Then the manifold $\R^{r + s}$ together with metric $\bar q^{\paren {r, s}}$, i.e. $\struct {\R^{r + s}, \bar q^{\paren {r, s}}}$, is called the '''pseudo-Euclidean space of signature $\tuple {r, s}$''' and is denoted by $\R^{r, s}$.
\end{definition}