Definition:Pseudo-Riemannian Metric
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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.
Source of Name
This entry was named for Georg Friedrich Bernhard Riemann.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics