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Let $M$ be a Riemannian manifold, $p\in M$, $T=T_pM$, and $\exp_p:T\to M$ the exponential map.

Is there a non-trivial bound from below on the maximum radius of a ball $B_r$ around the origin in $T$, such that $\exp_p(B_r)$ is still a geodesic space, that is, for any two points $x,y\in T$, a shortest path between $\exp_p(x)$ and $\exp_p(y)$ lies in $\exp_p(B_r)$?

For example, for the sphere of radius $R$, this is $r=\frac12\pi R$, while the injectivity radius is $\pi R$.

Similarly, I expect that the answer would involve the maximum curvature (supposing the curvature on $M$ is bounded from above).

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