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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 $X\langle Y,Z\rangle=0$?

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What have you tried until now? Where are you stuck? Why do not you take a look at Manfredo Perdigao do Carmo 'Riemannian Geometry'? There is a traitment of this point in §2.6 on Lie Groups on pages 40-41. Cf. for example here:… – Giuseppe Jun 2 '12 at 19:42
I don't know where to use the condition "left invariant". I read that chapter, but I still don't know the answer... – Jr. Jun 2 '12 at 19:54
up vote 6 down vote accepted

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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Why is $\langle dL_g Y_e, dL_g Z_e\rangle = \langle Y_e, Z_e\rangle$? – dmm Oct 9 '12 at 21:04
That's what it means for the metric to be biinvariant. – Eric O. Korman Oct 9 '12 at 22:26

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