Consider a Riemannian manifold $(M,g)$.
How much regularity is required of $g$ so that for any $x\in M$ and $v\in T\_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to M$ so that $\gamma(0)=x$ and $\dot\gamma(0)=v$?
All regularity is considered with respect to a fixed smooth (or somewhat less regular) atlas or (equivalently) in a fixed local coordinate chart.


If $g\in C^{1,1}$, then the Christoffel symbol is Lipschitz, whence existence and uniqueness follows from standard ODE theory.
For ODEs in general Lipschitz continuity is sufficient and necessary for existence and uniqueness.
It is not clear to me whether this is the case for geodesics, because the geodesic equation has a very specific structure and that may provide uniqueness in even lower regularity.


Existence requires less regularity than uniqueness, and [this old question](https://mathoverflow.net/q/133724/55893) is about existence for continuous metrics.
Since existence is known in very low regularity, my question concerns uniqueness (although I want to have both existence and uniqueness in as low regularity as possible).
If a metric can have a jump discontinuity, then both existence and uniqueness can be made fail, but I am not aware of any smoother counterexamples.


A refined (but equivalent) version my question is: *What are the best sufficient and necessary regularity conditions we know (whether or not they coincide in our current knowledge) on a Riemannian metric for **uniqueness** of geodesics?*
I am not aware of any positive or negative uniqueness results for $C^{0,\alpha}$ or $C^{1,\alpha}$ below $C^{1,1}$.
For example, are geodesics unique if $g\in C^{1,\alpha}$ for some $\alpha>0$, or are there perhaps counterexamples for all $\alpha<1$?
If you find the question unclear, please ask for details.


At least for $g\in C^1$ the geodesic equation is a well-defined classical ODE.
I guess that satisfying the geodesic equation and minimizing arc length locally are equivalent in this regularity, but I may be mistaken.



Existence and uniqueness of geodesics in low regularity
There is a classical example by Hartman that shows failure of uniqueness for $C^{1,\alpha}$ metrics. (P. Hartman, On the local uniqueness of geodesics, Amer. J. Math. 1950).


You could lower the regularity if the metric is smooth off some hypersurface but globally only $C^{0,1}$, see for example our recent review: [On geodesics in low regularity](http://iopscience.iop.org/article/10.1088/1742-6596/968/1/012010)


Moreover, being minimzing and solving the geodesic equations is not the same below $C^{1,1}$; see also Hartman or our review.


