---
tag: 5AKNNW3W4Z
kind: concept
name: Killing vector field
msc: [53B20, 53C20]
sources:
  - url: https://en.wikipedia.org/wiki/Killing_vector_field
same_as:
  wikidata: Q1200279
  wikipedia: Killing vector field
---

## Summary

A Killing vector field on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] is a vector field along which the metric does not change.
That is, the [[Lie derivative|PJWK0PKN6Q]] of the metric along it is zero.
Its local flow consists of [[isometries|R9YWQS15AA]], so Killing fields are the infinitesimal symmetries of the metric.
They form a Lie algebra, which is the Lie algebra of the isometry group when the manifold is complete.
In physics they give conserved quantities along geodesics.

## Explanations

### L1 human

Imagine sliding every point of a space a little along a vector field.
If the slide never stretches or shrinks any distance, the vector field is a Killing field.
On the round sphere, the field that turns everything around the north–south axis is a Killing field, since a rotation keeps distances.
In the plane, the constant field $(1, 0)$ is a Killing field, because translating the plane keeps distances, while the field $(x, y)$ is not, because its flow is a scaling.