Zoll sphere which is not a surface of revolution

Question

Are there examples of Zoll spheres which are not the surfaces of revolution?

Answer 1

You might need to be more precise in your question. As BS noted there exist Zoll $2$-spheres that are not surfaces of revolution, but you may have wanted to know whether there are any explicitly known examples. The answer to this is also 'yes'. For some examples, see Kiyohara, Two-dimensional geodesic flows having first integrals of higher degree, Math. Ann., 320 (2001), 487–505.

Answer 2

Yes, there are.

In fact, in this paper by LeBrun and Mason, they refer to a result by Guillemin 1976 according to which for any odd smooth function $f:S^2\to \mathbb{R}$, there is a one-parameter family $g_t=\exp(f_t)g_0$ of smooth Zoll metrics with $g_0$ the round one and $(df_t/dt)_{t=0}=f$.

If $f$ is not rotationally symmetric, you have examples.

Lebrun and Mason classify instead Zoll projective structures by very nice twistor-like constructions.

ADDED: Guillemin's result is detailed in A. Besse "Manifolds all of whose geodesics are closed", p.126 theorem 4.70.

Answer 3

I'm repeating what was said above and adding some details for completeness. A little chunk of historiography: Zoll found all the surfaces with periodic geodesic flow in the case of the isometry group isomorphic to $S^1$, i.e. the surfaces of revolution. It was done in his doctoral thesis in $1901$. First non-symmetrical results were found by Blaschke. The reference is W.Blaschke, Vorlesungen uber Differential geometrie, 1, Springer, Berlin, 1924.

By Korn-Licthenstein theorem (or simply by uniformization theorem) all the metrics on $S^2$ are conformal to a canonical one $g_0$. So any Riemanian metric $g$ on the sphere can be written in a form $g=\exp (\rho) g_0$. The space of Zoll metrics got people interested after the works of Zoll.

After Blaschke found the metrics which are not symmetrical by perturbation of Zoll's results, Hilbert asked the following question: what is a tangent space to the space of Zoll metrics at $g_0$? In other words, for what smooth functions on the sphere $\dot {\rho}$ do there exist Zoll deformations $g_t=\exp (\rho_t) g_0$ of the standart metric such that $\rho_0=-$ and $\frac{d \rho_t}{dt}=\dot{\rho}$ ar $t=0$? His doctoral student Funk gave a necessary condition: $\dot{\rho}$ should be odd with respect to antipodal map. Victor Guillemin shows that this is actually sufficient using Radon transform. The methods are powerful, the work is hard to read but great. The reference is The Radon Transofrm on Zoll surfaces, Advances in Mathematics, 22, 1976.