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March 2022 Closed geodesics on connected sums and $3$-manifolds
Hans-Bert Rademacher, Iskander A. Taimanov
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J. Differential Geom. 120(3): 557-573 (March 2022). DOI: 10.4310/jdg/1649953350


We study the asymptotics of the number $N(t)$ of geometrically distinct closed geodesics of length $\leq t$ of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups, and apply the results to the prime decomposition of a three-manifold. In particular we show that the function $N(t)$ grows at least like the prime numbers on a compact $3$-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact $3$-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simplyconnected manifold which is not homeomorphic to a sphere.


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Hans-Bert Rademacher. Iskander A. Taimanov. "Closed geodesics on connected sums and $3$-manifolds." J. Differential Geom. 120 (3) 557 - 573, March 2022.


Received: 12 September 2018; Accepted: 7 January 2020; Published: March 2022
First available in Project Euclid: 15 April 2022

Digital Object Identifier: 10.4310/jdg/1649953350

Keywords: closed geodesic , conjugacy classes , connected sum of manifolds , exponential growth , free product of groups , fundamental group

Rights: Copyright © 2022 Lehigh University


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Vol.120 • No. 3 • March 2022
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