Does every Zoll metric on $\mathbb{S}^2$ arise from a perturbation of the round metric?

Question

The introduction here states 'A formal perturbation argument of Funk later indicated that, modulo isometries and rescalings, the general Zoll metric on $\mathbb{S}^2$ depends on one odd function $f:\mathbb{S}^2\rightarrow\mathbb{R}$'. This implies that every Zoll metric on $\mathbb{S}^2$ depends on such a perturbation argument involving one odd function.

Page 2 of chapter 1 here states 'modulo isometries and rescalings, a general Zoll perturbation of the round metric on $\mathbb{S}^2$ depends on an odd function on $\mathbb{S}^2$'. This is a weaker statement, referring only to Zoll metrics arising from such a perturbation - there is no mention of the possibility of other metrics, which may not arise from a perturbation argument.

Could someone please confirm whether Zoll metrics on $\mathbb{S}^2$ have been completely classified by Funk's perturbation argument? Or are there other Zoll metrics on $\mathbb{S}^2$ that don't arise from such a perturbation?

Thanks!

Answer 1

The last I checked, it was unknown whether the set of Zoll metrics on the $2$-sphere was connected. What Guillemin (V. Guillemin, The Radon transform on Zoll surfaces, Advances in Mathematics 22 (1976), 85–119.) proved (roughly speaking) is that the set of Zoll metrics conformal to the round metric that are sufficiently near the round metric (in an appropriate sense of 'near') can be parametrized by a neighborhood of $0$ in the vector space of odd functions on the $2$-sphere. This says nothing about connectedness of the set of Zoll metrics on $S^2$ itself.