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?
Yes, sure. You change the definition of length, so you change the shortest paths between two given points, so change $\exp$.
(Edit: A nice visualization you get by looking at the upper half plane in $\mathbb{R}^2$. If this is equipped with the standard Euclidean metric, a geodesic is a straight line. If you use $g_{ij} = \frac{1}{y^2}\delta_{ij}$ you get one standard model of hyperbolic space, and the geodesics are half circles with center on the line $y=0$. See here)