---
tag: 9AHMM41PJ3
kind: concept
name: Jacobi field
msc: [53C22, 53C20]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobi_field
same_as:
  wikidata: Q1677714
  wikipedia: Jacobi field
---

## Summary

A Jacobi field is a vector field along a [[geodesic|NSYMQMHPDN]] that solves the Jacobi equation, a linear second-order equation involving the [[Riemann curvature tensor|JA29ETCM02]].
Jacobi fields are exactly the variation fields of families of geodesics, so they describe how nearby geodesics spread apart or come together.
They connect curvature with the behaviour of geodesics and lead to [[conjugate points|ZPXE5VSZ35]], comparison theorems and the second variation of length.

## Explanations

### L1 human

Take a family of geodesics that start close together, like lines of longitude leaving the north pole.
The small gap between neighbouring geodesics, as one moves along them, is a Jacobi field.
In a flat plane, such gaps grow at a constant rate, linearly.
On a sphere, positive curvature pulls the geodesics back together: the gap between two meridians grows like $\sin t$ and closes again at the south pole.