My native language is not English, so there may be many errors in my narration, but I will do my best to avoid them.I am currently learning to read do Carmo's Riemannian geometry. On pages 40 to 41.The author points out that
If G has a bi-invariant metric,productthatthethe innermetric determines on G satisfies the following relation: For any U,V,X $\in$$\mathcal{G}$,
$\left( 1 \right) \,\, \left< \left[ U,X \right] ,V \right> =-\left< U,\left[ V,X \right] \right> \,\,$
Author use some preliminary factsabout Lie groups.Please ask me to omit these tasks. Based on these facts,author say that
Let $\left< , \right>$ be a bi-invariantmetric on a Lie group G. Then for any X,U,V$\in$$\mathcal{G}$,$\left< U,V \right> =\left< dR_{x_t\left( e \right)}\circ dL_{{x_t}^{-1}\left( e \right)}U,dR_{x_t\left( e \right)}\circ dL_{{x_t}^{-1}\left( e \right)}V \right> =\left< dR_{x_t\left( e \right)}U,dR_{x_t\left( e \right)}V \right> $
Differentiating the expression above with respect to t, recalling that(,) is bilinear, and setting t = 0 in the expression obtained, weconclude that $$ 0=\left< \left[ U,X \right] ,V \right> +\left< U,\left[ V,X \right] \right> $$
So my question is the thirdth to last line in the final proof
Differentiating the expression above with respect to t
What is the feasibility of this operation ? In other words, based on what facts can we do this?
At the end of page 41, the author mentioned that using equation (1) to prove the characterization of bivariate measures on Lie group G is simple. Please allow my foolishness. I cannot complete this task. Can anyone help me.