Let $G$ be a Lie group with bi-invariant metric $\langle , \rangle$ and $X,Y,Z$ left invariant vector fields in $G$, how to conclude that $X\langle Y,Z\rangle=0$?
A question about left invariant vector fields
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$\begingroup$
lie-groups
riemannian-geometry
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0I don't know where to use the condition "left invariant". I read that chapter, but I still don't know the answer... – 2012-06-02
1 Answers
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The function $ G \ni g \mapsto \langle Y_g, Z_g \rangle $ is constant since $ \langle Y_g, Z_g\rangle = \langle dL_g Y_e, dL_g Z_e\rangle = \langle Y_e, Z_e\rangle. $ So differentiating it by $X$ gives 0. Note we have used the facts that $Y$ and $Z$ are left-invariant but $X$ need not be.
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1That's what it means for the metric to be biinvariant. – 2012-10-09