Let be $M$ a Riemannian manifold with Levi Civita connection. In the equation of geodesics appears the term $$\Gamma^k_{j} \circ \sigma(t) ,$$
where $\sigma :I \rightarrow M$ is a curve and $\Gamma^k_{j}$ are the Christoffel symbols..
If I change the parameter $t$ with $s=s(t)$, is it true that
$$\Gamma^k_{j} \circ \sigma(t)=\Gamma^k_{j} \circ \sigma(t(s))?$$
How can I show it?
EDIT Is this argument correct?
I know that using the Levi Civita connection the Christoffel symbols depend only on the $g_{\alpha,\beta}$, and I show that them are invariant under reparameterization : $$ g_{\alpha,\beta}(\sigma(t))= \langle \frac{\partial}{\partial x^\alpha},\frac{\partial}{\partial x^\beta}\rangle_{\big|\sigma(t)} = \langle \frac{\partial}{\partial x^\alpha},\frac{\partial}{\partial x^\beta}\rangle_{\big|\sigma(t(s))} ,$$ since the point is the same.