TITLE: On which closed Riemannian manifolds are geodesics always recurrent?
QUESTION [5 upvotes]: Let $M$ be a closed Riemannian manifold. What are the necessary and sufficient conditions on $M$ to ensure that for every point $p \in M$, and every geodesic $\gamma: [0, \infty) \to M$ with $\gamma(0) = p$, we have that $\liminf_{t \to \infty} d(\gamma(t), p) = 0$?
Note: Here d denotes the usual Riemannian distance.

REPLY [3 votes]: Just so that it doesn't get lost in the comments: in this paper Nadirashvili shows that a a $C^2$ Riemannian metric in the 2-torus for which the geodesic flow is recurrent (all points are Poisson stable) must be flat.
N. S. Nadirashvili, “Conditions of stability in the sense of Poisson of a geodesic flow on a torus”, Mat. Zametki, 44:1 (1988),  147–149