Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the lengths of closed geodesics on a two-sphere

Author(s): Nancy Hingston
Journal: Proc. Amer. Math. Soc. 125 (1997), 3099-3106.
MSC (1991): Primary 58E10; Secondary 53C22
MathSciNet review: 1443831
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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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