On the lengths of closed geodesics on a two-sphere

Hingston, Nancy

doi:10.1090/S0002-9939-97-04235-4

On the lengths of closed geodesics on a two-sphere
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Proc. Amer. Math. Soc. 125 (1997), 3099-3106 Request permission

Abstract:

Let $c$ be an isolated closed geodesic of length $L$ on a compact Riemannian manifold $M$ which is homologically visible in the dimension of its index, and for which the index of the iterates has the maximal possible growth rate. We show that $M$ has a sequence $\{c_n\}$, $n\in \mathbb {Z}^+$, of prime closed geodesics of length $m_nL-\varepsilon _n$ where $m_n\in \mathbb {Z}$ and $\varepsilon _n\downarrow 0$. The hypotheses hold in particular when $M$ is a two-sphere and the “shortest” Lusternik-Schnirelmann closed geodesic $c$ is isolated and “nonrotating”.

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