# Definition:Pseudo-Riemannian Metric

Jump to navigation
Jump to search

## 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