The Berger conjecture for three dimensional manifolds
Abstract
A conjecture of Berger says that on a simply connected manifold all of whose geodesics are closed, all unit speed geodesics have the same shortest period. In this paper we will show that the S^1-action induced by the geodesic flow is free. We use symplectic geometry to investigate the topology of the quotient of the unit tangent bundle T^1M/S^1 by the S^1 action that comes from the geodesic flow. We show that this quotient admits the structure of a smooth symplectic manifold and that this symplectic manifold is symplectomorphic to S^2 \times S^2 equipped with a split form. This will enable a computation involving the Euler class of the orbibundle T^1M\rightarrow T^1M/S^1 which will lead to a solution of Berger's conjecture for three-dimensional manifolds.
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BibTeX
@misc{dittmer2026berger,
title = {{The Berger conjecture for three dimensional manifolds}},
author = {Jakob Dittmer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41006},
eprint = {2610.41006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41006},
}
RIS
TY - UNPB
TI - The Berger conjecture for three dimensional manifolds
AU - Jakob Dittmer
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.41006
UR - https://arxiv.org/abs/2610.41006
AB - A conjecture of Berger says that on a simply connected manifold all of whose geodesics are closed, all unit speed geodesics have the same shortest period. In this paper we will show that the S^1-action induced by the geodesic flow is free. We use symplectic geometry to investigate the topology of the quotient of the unit tangent bundle T^1M/S^1 by the S^1 action that comes from the geodesic flow. We show that this quotient admits the structure of a smooth symplectic manifold and that this symplectic manifold is symplectomorphic to S^2 \times S^2 equipped with a split form. This will enable a computation involving the Euler class of the orbibundle T^1M\rightarrow T^1M/S^1 which will lead to a solution of Berger's conjecture for three-dimensional manifolds.
ER -