Hello fellow mathematicians
i am looking for the proof of existence and uniqueness of geodesics in a random differential manifold. I know that this can be demostrated using Differential equations Theory.
Summarizing, we have the following System of differential equations:
$$\ddot{x}^i+\Gamma _{nm}^{i}\dot{x}^n\dot{x}^m=0,$$
where $x^i$ is the composition of the curve with the i-component of the local coordinate system. This can be simplifice as
$$\dot{v}^i+\Gamma _{nm}^{i}(x)v^nv^m=0,$$ $$\dot{x}^i=v^i.$$ I know that we have to apply Picard-Lindelöf Theorem of existence and uniqueness. Nevertheless, i dont find any source where the proof is developed in detail.
I'm more interesting in the proof of the Lipschitz's condition, i supose that we have to use the properties of Christoffel symbols.
I would need some details of the demostration or any source where i could read the complete proof.
thanks in advance.