Given a smooth manifold $M$ (Hausdorff,second countable), and let $g$ be a pseudo-Riemannian metric but is only $C^1$ (or $C^k$ or continuous).
Can we induce a Levi-Civita connection?
Can we talk about geodesics and exponential map, what about the regularities?
Thank you in advance.
On
This question is awfully broad, but I can give a little summary.
If $g\in C^{1,1}$ (which is a subset of $C^2$), then the geodesic equation has unique solutions for any initial point and direction. This follows because in that case you have a system of equations with Lipschitz regularity, which is enough for existence and uniqueness. The Christoffel symbol exists and has regularity $C^{k-1}$ if $g\in C^k$; it contains first order derivatives of the metric.
For general ODE theory Lipschitz regularity is the threshold for uniqueness, but the geodesic equations have special structure. It's not clear whether the solutions are unique for $g\in C^{1,\alpha}$ or $g\in C^1$ even in the Riemannian setting. There is an unanswered MathOverflow question on this problem.
Once you have existence and uniqueness for geodesics, you can define exponential maps and all that.
The regularity of geodesics and that of the exponential map may depend on the regularity of your manifold structure. If $g\in C^k$, I would expect that a geodesic has regularity $C^{k+1}$ based on the geodesic equation. But if the manifold only has $C^k$ structure, this doesn't mean anything, so you may come down to $C^k$.
The proof of the existence of the Levi Civita connection depends only of the first derivative , so it can be defined for $C^k$ differentiable metrics, $k\geq 1$.
https://en.m.wikipedia.org/wiki/Levi-Civita_connection