Definition:Pseudo-Euclidean Metric

Definition

Let $\struct {M, g}$ be a pseudo-Riemannian manifold.

Let $\tuple {\xi^1, \dotsc, \xi^r, \tau^1, \dotsc, \tau^s}$ be the standard coordinates of $\R^{r + s}$.

The pseudo-Euclidean metric (of signature $\tuple {r, s}$) is the pseudo-Riemannian metric, which in the standard coordinates reads:

$g = \paren {\d \xi^1}^2 + \dotsb + \paren {\d \xi^r}^2 - \paren {\d \tau^1}^2 - \dotsb - \paren {\d \tau^s}^2$

Sources

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