This may be a silly question but I am new to Riemannian Geometry.
If I have two different Riemannian Metrics $g_1,g_2$ on a smooth manifold $M$, then do the geodesics on the Riemannian manifolds $(M,g_1)$ and $(M,g_2)$ differ? That is do the Exponential maps depend on our choice of Riemannian metric?