Let $(\mathbb{S}^2,g)$ be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and have the same length $L > 0$, say.


**Question 1**: in the celebrated book of Besse entitled "Manifolds all of whose geodesics are closed", the author asserts that every minimizing geodesic $\gamma$ from $p \in \mathbb{S}^2$ to an arbitrary point $q \in \mathbb{S}^2$ has length less than $L/2$, and thus the diameter of $(\mathbb{S}^2,g)$ is less than $L$. How does one prove these claims?


**Question 2**: In the paper by Bott entitled "On manifolds all of whose geodesics are closed", the author defines the index of $(\mathbb{S}^2,g)$ (or for any manifold of higher dimension with this property on the geodesics) as the index of any closed geodesic in the surface. Why is this number independent of the geodesic and of its parametrization?



On properties of Besse spheres
It so happens I recently read this same passage in Besse's book. I am no expert, but here is how I understand it.


**Question 1.** Let $p,q \in \mathbf{S}^2$ be two arbitrary points, and $\eta: [0,L] \to \mathbf{S}^2$ be a closed geodesic containing both points, fixing $\eta(0) = p$ for example. Going along $\eta$ either in the given direction $\eta'(0)$, or in the opposite direction $-\eta'(0)$, one encounters $q$ after a distance $L/2$ at the most. Therefore $d(p,q) \leq L/2$, and as $p,q \in \mathbf{S}^2$ were arbitrary, the diameter of $\mathbf{S}^2$ is also at most $L/2$.


*Remark.* Besse gets a bound of $L$ for the diameter, because the situation there is slightly different: they work with manifolds for which all geodesics issued from a (fixed) point $p$ are loops with length $L$.


**Question 2.** The second question is a bit trickier; to answer it I will still use Besse's book. There the constancy of the index is stated as a part of Theorem 7.23, the Bott–Samelson theorem. Note that technically Bott states that all geodesics issued *from the same point $p$* have the same index; this is also how it's stated in Besse's book, so I'll focus on this claim.


*Claim.* The geodesics of length $L$ issued from $p \in \mathbf{S}^2$ have the same index.


We write $\lambda(\gamma) \in \mathbf{Z}$ be the index of any closed geodesic $\gamma: [0,L] \to \mathbf{S}^2$ with $\gamma(0) = p = \gamma(L)$. (We consider the index as a geodesic arc with fixed endpoints, rather than as a geodesic loop.) The Morse index theorem states that $\lambda(\gamma)$ equals the number of conjugate points $\gamma(t)$ for all times $0 < t < L$, counted with multiplicity.


Let $(\gamma\_s \mid -\delta < s < \delta)$ be some continuous variation of $\gamma\_0 := \gamma$ through geodesic loops at $p$. The times at which one encounters the first, second, $\dots,n$th conjugate points along $\gamma\_s$ vary continuously with $s$. The only way the index $\lambda(\gamma\_s)$ could change would be if conjugate points either appeared in the arc $\gamma\_s$ or disappeared from it. In either case, some conjugate points would need to cross the base point $p$, adding to the multiplicity of $p$ along the arc $\gamma\_{s^\*}$ for some $s^\* \in (- \delta,\delta)$.


This is impossible because the multiplicity of $p$ to itself along any geodesic loop is already 'maxed out': because the Jacobi fields are $L$-periodic, the point $p$ is conjugate to itself with multiplicity one. (This is the largest possible multiplicity on a two-dimensional surface.) Therefore the loops issued from $p$ all have the same index, which finishes the proof of the claim.


