A theorem of Bott 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.

Manifolds all of whose geodesics are closed, Ergeb. Math., vol.91, 1976, 250 pp.) $\endgroup$ – Vesselin Dimitrov Jan 9 '18 at 18:20simple. $\endgroup$ – Sam Nead Jan 9 '18 at 18:54