It is known that isometric metrics have the same exponential maps. I am interesting in the converse, can we recover the metric from the exponential map?
Suppose that $(M,g_1)$ and $(M,g_2)$ are Riemannian manifolds with the same exponential maps, i.e., the maps $\exp_{1}:TM\rightarrow M$ and $\exp_{1}:TM\rightarrow M$ are equal. Does it follow that $(M,g_1)$ is isometric to $(M,g_2)$?
If someone have already asked this question I will appreciate the answer.