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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?

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    $\begingroup$ See the discussion on the bottom of p.2 of arxiv.org/pdf/2401.03417.pdf. The setting there is slightly different. $\endgroup$
    – Igor Belegradek
    Commented Mar 16 at 11:50
  • $\begingroup$ Thank you so much for the reference! This preprint says that the desired regularity holds for submanifolds in Euclidean space (using the Gauss equation to get smoother curvature tensor). It'd be nice to know whether it also holds true for general manifolds. Nevertheless, there are some interesting ideas and useful references in that arXiv preprint. $\endgroup$
    – Sean
    Commented Mar 17 at 2:16
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    $\begingroup$ I suspect the proof can be adapted to $C^{2,\alpha}$ Riemannian manifolds. I am puzzled why the paper is restricted to submanifolds of $\mathbb R^n$. The smoothing with bounded curvature already works for $C^{1,1}$ metrics, and the argument in the paper should imply that geodesic flows converge. $\endgroup$
    – Igor Belegradek
    Commented Mar 17 at 2:39

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