---
tag: 7CBT4SM83Y
kind: concept
name: Riemannian manifold
msc: [53C20, 53B20]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Riemannian_manifold
same_as:
  wikidata: Q632814
  wikipedia: Riemannian manifold
---

## Summary

A Riemannian manifold is a smooth manifold together with a [[Riemannian metric|Q61VFHCH2J]].
It is the setting of Riemannian geometry, where one studies lengths, angles, geodesics, volume and curvature on curved spaces.
Euclidean space, spheres and hyperbolic spaces are the basic examples, and every submanifold of Euclidean space inherits a Riemannian structure.
Replacing positive definiteness by a Lorentzian signature gives the [[pseudo-Riemannian manifolds|GH6CXWFNDJ]] that model spacetime in general relativity.

## Explanations

### L1 human

A Riemannian manifold is a curved space where you can measure.
Near each point it looks like a small piece of ordinary space, and you know how to measure lengths and angles there.
From this you can find the shortest paths, compute areas, and detect how curved the space is.
The surface of the Earth, with distances measured along the surface, is a two-dimensional example.