Proving del_X(Y)=0.5[X,Y] in Lie Group Geometry

[sroeyz]

Hello,
I seem to be having difficulty proving something.
I hope you can help me.

I will write del_X(Y) when I refer to the levi-chivita connection (used on Y in the direction of X).

Let G be a lie group, with a bi-invariant metric , g , on G.
I want to prove that del_X(Y) = 0.5 [X,Y] (Lie brackets) , whenever X,Y are left-invariant vector fields on G.

Thanks in advance.

 

[Reb]

Hello,
I seem to be having difficulty proving something.
I hope you can help me.

I will write del_X(Y) when I refer to the levi-chivita connection (used on Y in the direction of X).

Let G be a lie group, with a bi-invariant metric , g , on G.
I want to prove that del_X(Y) = 0.5 [X,Y] (Lie brackets) , whenever X,Y are left-invariant vector fields on G.

Thanks in advance.

The Levi-Civita connection can be expressed via the inner product (which is a fundamental result in metric differential geometry). Use this formula and bi-invariance to obtain the result.