Let $(M,g)$ be a Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold. The zero section is denoted by $Z$. We define a Hamiltonian on $T^0 M=TM\setminus Z$ via $$H=\frac{Ric(V,V)}{|V|^2}$$ For Einstein manifold this produce the trivial Hamiltonian dynamics(All points are singularity). But what about general case? What can be said about the critical points of this Hamiltonian?
In particular is there an example of a manifold not admitting any Einstein structure but it admits a metric whose above constructed Hamiltonian satisfies the following property: All solutions are periodic or singularity. (Or all solutions remains in compact sets)
Note: One can go ahead and ask some questions as follows: For what kind of manifolds the geodesic flow is smoothly or topological equivalents to the Hamiltonian flow associated to the Hamiltonian $Ric(V,V)$? Can the ergodic properties of this Hamiltonian flow be determined in terms of the sign of curvature? Under what geometric conditions the two Hamiltonian $H_1=|V|^2$ and $H_2=Ric(V,V)$ commute wrt Poisson bracket,$\{H_1,H_2\}=0$?
