2016/08/05 by Rademacher, Hans-Bert
#53C22 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1608.01937
The existence of two geometrically distinct closed geodesics on an n-dimensional sphere Sn 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 ∈ ℕ all closed geodesics of index ≤ N of a non-reversible and bumpy Finsler metric on Sn are geometrically equivalent to the closed geodesic c then there is a covering cr of minimal index growth, i.e. \rm ind(crm)=m \rm ind(cr)-(m-1)(n-1) for all m ≥ 1 with \rm ind(crm)≤ N. But this leads to a contradiction for N =∞ 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 S2 a metric of positive flag curvature carrying only two closed geodesics of length