I would like to ask the community for a reference on the following question:
Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that the orbits of the geodesic flow $\varphi:T^1M\longrightarrow T^1M$ are geodesics of $(T^1M,g_S)$?
Any help/reference would be really appreciated!