[A theorem of Bott](https://www.jstor.org/stable/1969839) states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.


The problem of constructing such metrics is known to be extremely hard.


So my particular interest is: what else is known about this problem in the case of dimension $3$?


For example, are all homology spheres known to admit such metrics? If a homology sphere (or a homology $\mathbb{RP}^3$ --- by the way, are there any examples which are not $\mathbb{RP}^3$?) has finite $\pi\_1$, its universal cover is $S^3$, and I believe that all geodesics will be closed in the metric induced by the standard metric on $S^3$. 


But I've just learned that there are examples of homology spheres with infinite fundamental group. Their Thurston geometry is modeled on the universal cover of $\operatorname{SL}(2, \mathbb{R})$. It would be very interesting to know if such homology spheres admit metrics with all geodesics closed.



3-manifolds with all geodesics closed
There are integral homology spheres in the following Thurston geometries: $S^3$, $\mathrm{PSL}(2,\mathbb{R})$, and $H^3$. No manifold of the latter two types (when equipped with any metric) can have all of its geodesics being closed. This follows from observing that the fundamental group contains a non-trivial free group, and then a Gromov-Hausdorff limiting argument.


