4
$\begingroup$

I want to find the integral curves of $[X,Y]$, then maybe can use this to prove. Can anyone gives an answer ? Thanks.

  • 0
    Just prove if $X$,$Y$ are both killing vector fields,then $[X,Y]$ is also killing vector field.Can anyone give some suggestions?Thanks.2012-03-14

1 Answers 1

6

From the Jacobi identity and definition of Lie derivative it follows that $L_{[X,Y]} = L_X \circ L_Y - L_Y \circ L_X$. Thus $L_{[X,Y]} g = 0$ if $L_Y g = 0$ and $L_X g = 0$.