I've done the following reflection, but I don't know it it is correct or not:
Let $G=g_{ij}$ the metric tensor of a differentiable manifold $M$, then we could consider it as a map: $G:TM\rightarrow\mathbb{R} $ which for $p\in M$ and $v\in T_p$ associate $G(v)=g_{ij}v^iv^j$.
If we take the differential of $G$, $dG\in \Omega (TM)$, it is a differential form over $TM$, so in general it admits integral curves $\gamma_s(t):I\rightarrow TM$, parametrised by $s$.
So, for a given point of $M$ and its tangent vector, we could look for a $s$ such that $\gamma_s$ passes for it. If we take the composition of $\gamma_s$ and the projection map $\pi:TT\rightarrow M$, we obtain a curve of $M$.
My question is: are these curves the geodesics?