---
tag: 1VM1AQT956
kind: concept
name: Space form
msc: [53C20, 53C21]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Space_form
same_as:
  wikidata: Q4381552
  wikipedia: Space form
---

## Summary

A space form is a complete connected [[Riemannian manifold|7CBT4SM83Y]] of constant [[sectional curvature|1X42JAA349]].
The three simply connected models are the round sphere, Euclidean space and hyperbolic space.
Every space form is a quotient of one of these models by a group of isometries.
So space forms are the simplest and most symmetric Riemannian manifolds.
They serve as the reference spaces in comparison geometry.

## Explanations

### L1 human

A space form is a space that curves by the same amount at every point and in every direction.
If the amount is positive, the space looks like a sphere; if it is zero, it looks flat.
If it is negative, it looks like the hyperbolic plane, a saddle everywhere.
A flat torus, made by gluing opposite sides of a square, is a space form of curvature zero.
An open disc in the plane is flat but has an edge one can reach in finite distance, so it is not complete and not a space form.