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--Long [7] and the author [27]. 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. $${\rm ind}(c^{rm})=m {\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \ge 1$ with ${\rm ind}\left(c^{rm}\right)\le N.$ But this leads to a contradiction for $N =\infty$ as pointed out by Goresky--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.