---
tag: GH6CXWFNDJ
kind: concept
name: Pseudo-Riemannian manifold
msc: [53B30, 53C50]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold
same_as:
  wikidata: Q1510587
  wikipedia: Pseudo-Riemannian manifold
---

## Summary

A pseudo-Riemannian manifold is a [[smooth manifold|F45VQ47GPV]] with a smoothly varying [[nondegenerate|01NHFZF2SK]] [[symmetric bilinear form|0AXZNE5XNH]] on each tangent space, which need not be positive definite.
It generalises a [[Riemannian manifold|7CBT4SM83Y]], where the form is positive definite.
The most important case is a [[Lorentzian manifold|BY4QYYS788]], the setting of general relativity.
Much of Riemannian geometry, such as the [[Levi-Civita connection|CAZYGE1T4E]], geodesics and curvature, works in this setting.

## Explanations

### L1 human

On a Riemannian manifold every nonzero vector has a positive squared length.
A pseudo-Riemannian manifold drops this rule: some vectors may have negative or zero squared length.
What is kept is that the form has no hidden directions: no nonzero vector is orthogonal to every vector.
Minkowski space $\R^4$ with $-dt^2 + dx^2 + dy^2 + dz^2$ is the basic example.
There, a vector such as $(1, 1, 0, 0)$ has squared length $0$ although it is not zero.