# Under exponential map in a Riemannian manifold, what radius keeps the image of a ball a geodesic space?

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).