If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an
Isometry), then the Vector field is called a Killing vector.

(1) 
so let

(2) 

(3) 
where is the Lie Derivative. An ordinary derivative can be replaced with a covariant derivative in a
Lie Derivative, so we can take as the definition

(5) 

(6) 
which gives Killing's Equation

(7) 
A Killing vector satisfies

(8) 

(9) 

(10) 
where is the Ricci Tensor and is the Riemann Tensor.
A 2sphere with Metric

(11) 
has three Killing vectors, given by the angular momentum operators
The Killing vectors in Euclidean 3space are



(15) 



(16) 



(17) 



(18) 



(19) 



(20) 
In Minkowski Space, there are 10 Killing vectors
The first group is Translation, the second Rotation, and the final corresponds to a
``boost .''
© 19969 Eric W. Weisstein
19990526