Why there is a minus sign in the Killing's equation

[maxverywell]

With the Levi-Civita connection, the Killing's equation is:
$\xi_{a;b}+\xi_{b;a}=\xi_{a,b}+\xi_{b,a}-2{\Gamma^{c}}_{ab}\xi_{c}=0$

I can't understand why there is a minus sign in front of the Christoffel symbols.

We have that:
$\xi_{a;b}=\xi_{a,b}+{\Gamma^{c}}_{ab}\xi_{c}$
$\xi_{b;a}=\xi_{b,a}+{\Gamma^{c}}_{ba}\xi_{c}$
and because of ${\Gamma^{c}}_{ab}={\Gamma^{c}}_{ba}$, it should be
$\xi_{a;b}+\xi_{b;a}=\xi_{a,b}+\xi_{b,a}+2{\Gamma^{c}}_{ab}\xi_{c}=0$
and not with a minus sign.

 

[sweet springs]

Hi

For tesor in variant components

$\xi_{a;b}=\xi_{a,b}-{\Gamma^{c}}_{ab}\xi_{c}$

 

[maxverywell]

Oops, right, [tex]\xi_a[/itex] has lower index i.e. it's a component of covector and its covariant derivative has a minus sign in front of the connection coefficients. Thnx!