Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies?
Since $\mathrm{exp}_p$ is defined by the geodesic equation, the question relates smooth dependence of the geodesic equation $$ \frac{d^2 f^s}{dr^2} + \Gamma^s_{ij} \frac{d f^i}{dr} \frac{d f^j}{dr}=0 $$ where the Christoffel symbol $\Gamma^s_{ij}\in C^{1,\alpha}$.
I understand that from a general ODE theory, if the coefficient is Lipschitz continuous, then the solution is Lipschitz continuous in the initial conditions (while the smooth dependence fails for $C^{0,\alpha}$ coefficients). Thus, one can conclude Lipschitz continuous of $\mathrm{exp}_p$ in $p$. Can it actually be $C^{1,\alpha}$ though?
