# Is this covariant derivative identity true?

Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$

If this is true, I'm making progress. If it isn't, it's back to the drawing board. Does anyone know if it is correct? I've tried writing it out as$$\left(\frac{\partial u^{\alpha}}{\partial x^{\beta}}+u{}^{\gamma}\Gamma_{\gamma\beta}^{\alpha}\right)u_{\alpha}=\left(\frac{\partial u_{\alpha}}{\partial x^{\beta}}-u_{\gamma}\Gamma_{\alpha\beta}^{\gamma}\right)u^{\alpha},$$ but I'm none the wiser.

• It is true assuming that the connection is the Levi-Civita one, as it is a metric'' connection... – Valter Moretti Nov 17 '14 at 19:31
• For TeX purposes, you can use \, to create a space: $u^\alpha_{\,;\beta}$ creates $u^\alpha_{\,;\beta}$. – Kyle Kanos Nov 17 '14 at 19:38
• @ValterMoretti Any hints as to how I can show it's true? – Peter4075 Nov 17 '14 at 19:48