---
tag: Q61VFHCH2J
kind: concept
name: Riemannian metric
msc: [53B20, 53C20]
specializes: [WX302T93TW]
sources:
  - url: https://en.wikipedia.org/wiki/Metric_tensor
---

## Summary

A Riemannian metric on a smooth manifold is a smoothly varying [[inner product|0NB0AWGJBW]] on each tangent space.
It lets one measure lengths of tangent vectors and angles between them, and from these the lengths of curves, distances, areas and volumes.
It is a symmetric positive definite [[tensor field|WX302T93TW]] of type $(0, 2)$.
Every smooth manifold admits Riemannian metrics, and the choice of one is the starting point of Riemannian geometry.

## Explanations

### L1 human

On a curved surface, a small piece looks almost flat, and on a flat piece you can use a ruler and a protractor.
A Riemannian metric gives such a ruler at every point, possibly different from point to point.
To measure a curve, add up the lengths of its tiny pieces with the local ruler.
In the upper half-plane, the metric $(dx^2 + dy^2)/y^2$ makes rulers shrink as you approach the $x$-axis, so the axis is infinitely far away.
This is the hyperbolic plane.