---
tag: 97NYHNBZ2D
kind: concept
name: Sub-Riemannian manifold
msc: [53C17]
sources:
  - url: https://en.wikipedia.org/wiki/Sub-Riemannian_manifold
same_as:
  wikidata: Q7630587
  wikipedia: Sub-Riemannian manifold
---

## Summary

A sub-Riemannian manifold is a [[smooth manifold|F45VQ47GPV]] where motion is allowed only along chosen horizontal directions.
These form a subbundle of the tangent bundle, and they carry an inner product.
Distance is the infimum of lengths of curves that are tangent to the horizontal directions at every time.
Assume that the horizontal directions and their [[Lie brackets|68FGQEBD6J]] span all directions.
Then, by the Chow–Rashevskii theorem, any two points of a connected manifold are joined by a horizontal curve, and the distance is finite.
Sub-Riemannian geometry appears in control theory, in the study of hypoelliptic operators and in the geometry of the Heisenberg group.

## Explanations

### L1 human

Think of a car: it can drive forward and backward and turn its wheels, but it cannot move sideways.
Still, by combining these moves, as in parallel parking, it can reach every position.
A sub-Riemannian manifold models this: only some directions of motion are allowed, but combining them reaches everywhere.
The distance between two places is then the length of the shortest allowed route, which can be much longer than in the unrestricted space.