Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
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: books.google.it/… –  Giuseppe Tortorella 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

1 Answer 1

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.

share|improve this answer
    
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
1  
That's what it means for the metric to be biinvariant. –  Eric O. Korman Oct 9 '12 at 22:26

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.