---
tag: PAA93R4WYD
kind: concept
name: Symmetric space
msc: [53C35]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_space
same_as:
  wikidata: Q3058244
  wikipedia: Symmetric space
---

## Summary

A symmetric space is a Riemannian manifold that has, at each point, an isometry reversing all geodesics through that point.
Euclidean spaces, round spheres, hyperbolic spaces and projective spaces are symmetric spaces.
Each symmetric space is a quotient $G/K$ of a Lie group by a closed subgroup, so its geometry reduces to Lie theory.
[[Cartan's classification|9P8VK3H4YZ]] lists all of them, which makes them a large and well understood family of model spaces.

## Explanations

### L1 human

Stand at a point of the space and look in some direction.
In a symmetric space there is a symmetry of the whole space that keeps you fixed and swaps every direction with its opposite.
On a round sphere, this is the rotation by a half turn about the axis through you and the antipodal point.
So the space looks the same from every point and in every pair of opposite directions.