# 3-manifolds with all geodesics closed

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.

• Just to be sure: I assume you are aware of Arthur Besse's (old) extensive monograph on this subject? (Manifolds all of whose geodesics are closed, Ergeb. Math., vol. 91, 1976, 250 pp.) Jan 9, 2018 at 18:20
• A nice quotient of the round three-sphere $S^3$ (say, a lens space) will have all geodesics closed, but need not have the homology of $S^3$ or of $P^3$. So I don't understand the statement that you attribute to Bott. Jan 9, 2018 at 18:48
• Ah - indeed, you have left out a hypothesis. Bott is assuming that all geodesics emanating from some special point $p$ are closed and simple. Jan 9, 2018 at 18:54
• The connected sum of $\mathbb{RP}^3$ with a non-trivial homology sphere is a non-trivial homology $\mathbb{RP}^3$. Jan 9, 2018 at 21:13

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.