---
tag: R9YWQS15AA
kind: concept
name: Isometry of Riemannian manifolds
msc: [53C20, 53B20]
specializes: [02HPP5VCZH]
sources:
  - url: https://en.wikipedia.org/wiki/Isometry
---

## Summary

An isometry of Riemannian manifolds is a [[diffeomorphism|0KZZ70ZHXP]] that carries one [[Riemannian metric|Q61VFHCH2J]] to the other.
Two isometric manifolds are the same for Riemannian geometry: lengths, angles, geodesics and curvature correspond.
The isometries of a manifold onto itself form a group, which by the [[Myers–Steenrod theorem|Z3VRMW8ETQ]] is a Lie group.

## Explanations

### L1 human

An isometry moves a curved space onto another without stretching or shrinking anything, even infinitesimally.
Rotations of the round sphere are isometries of the sphere onto itself.
Rolling a flat sheet of paper into a cylinder is locally an isometry, because the paper is bent but not stretched.
A map from the sphere to the plane can never be an isometry, even on a small piece, because the sphere is curved and the plane is flat.