Bumpy metrics on spheres and minimal index growth

Abstract

The existence of two geometrically distinct closed geodesics on an n-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [7] and the author [25]. We simplify the proof of this statement by the following observation: If for some $N \in \mathbb {N}$ all closed geodesics of index $\le$N of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic c, then there is a covering $c^r$ of minimal index growth, i.e.,

$$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$

for all $m \ge 1$ with $\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.$ But this leads to a contradiction for $N =\infty$ as pointed out by Goresky and Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large $L>0$, we obtain on $S^2$ a metric of positive flag curvature carrying only two closed geodesics of length $<L$ which do not intersect.