Is the Sum of Christoffel Symbols Equal to Their Negative in Tensor Calculus?

[elfmotat]

Is the following true?
\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha}=-2\Gamma_{\alpha \mu \nu}
where:
\Gamma_{\alpha \mu \nu}=g_{\alpha \sigma}\Gamma^{\sigma}_{~\mu \nu}

I ask because, while bored in a philosophy lecture, I decided to try to derive the geodesic equation by extremizing ∫g_μνu^μu^νdλ, where u^μ = dx^μ/dλ.

I was able to arrive at the following, where a_μ=du_μ/dλ:
2a_\alpha = (\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha})u^\mu u^\nu

So, am I on the right track or did I make an error somewhere?

 

[elfmotat]

Nevermind, they are clearly not equal. From the definition of the Christoffel symbols in terms of the metric, I found that:

(\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha})=\partial_\alpha <br /> g_{\mu \nu }
This makes sense, because \nabla_\alpha g_{\mu \nu }=0.

Unfortunately for me though, this is clearly not equal to -2\Gamma_{\alpha \mu \nu} given that:

-2\Gamma_{\alpha \mu \nu}=\partial_\alpha g_{\mu \nu}-\partial_\mu g_{\nu \alpha}-\partial_\nu g_{\mu \alpha }