# nLab Killing vector

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

A Killing vector on a (pseudo-)Riemannian manifold is a covariantly constant vector : a vector field $v \in \Gamma(T C)$ that is annihilated by (the symmetrization of) the covariant derivative of the corresponding Levi-Civita connection.

The flows of Killing vectors are isometries of the Riemannian manifold onto itself.

Similarly a Killing spinor is a covariantly constant spinor.

Revised on September 17, 2011 10:48:39 by Urs Schreiber (89.204.137.81)