Definition:Pseudo-Euclidean Space of Signature (r,s)

Definition

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 $\R^{r, s}$.

Also defined as

Whether $\xi$ or $\tau$ is related to pluses or minuses depends on the source.

However, this amounts to the multiplication of the metric by $-1$, so derived results are affected trivially.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics