I want to find the integral curves of $[X,Y]$, then maybe can use this to prove. Can anyone gives an answer ? Thanks.
Prove that Killing vector fields form Lie algebra.
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differential-geometry
riemannian-geometry
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0Just 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
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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 $L_X g = 0$.