---
tag: B4QR306D8G
kind: concept
name: Riemannian distance
msc: [53C20, 53B20]
sources:
  - url: https://en.wikipedia.org/wiki/Riemannian_manifold
---

## Summary

The Riemannian distance between two points of a connected [[Riemannian manifold|7CBT4SM83Y]] is the infimum of the lengths of the curves that join them.
It turns the manifold into a [[metric space|0VKD1TP7F3]] whose topology is the manifold topology.
This distance is the bridge between the infinitesimal data of the [[Riemannian metric|Q61VFHCH2J]] and global questions such as completeness, diameter and shortest paths.

## Explanations

### L1 human

A Riemannian metric tells how long a tiny step is at each point.
Adding up the tiny steps gives the length of a whole path.
The distance between two points is the length of the shortest way between them, or the limit of shorter and shorter ways when no shortest one exists.
On the round unit sphere, the distance between the north pole and a point on the equator is $\pi/2$, the length of a quarter of a great circle.
Remove one point from the plane, and take two points on opposite sides of the hole.
Their distance is the length of the straight segment, but no path in the punctured plane has this length.