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Hopf conjecture holds for analytic, k-basic Finsler tori without conjugate points

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Abstract

We show that analytic, k-basic Finsler metrics in the two torus without conjugate points are analytically integrable, in the sense that the unit tangent bundle of the metric admits an analytic foliation by invariant Lagrangian graphs. This result, combined with the fact that C 1,L integrable k-basic Finsler metrics in the two torus have zero flag curvature (Barbosa-Ruggiero [19]) implies that analytic k-basic Finsler metrics in two tori without conjugate points are flat, a positive answer to the so-called Hopf conjecture for tori without conjugate points. Since there are well known examples of non flat tori without conjugate points (Busemann was the first to show such examples) the Hopf conjecture is not true if we drop the k-basic assumption. As for higher dimensional tori, a quite simple argument based on Schur’s Lemma shows that the only Finsler, k-basic (3 + m)-tori are the flat ones for every m ≥ 0.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Juiz de Fora, 36036-900, Juiz de Fora, MG, BRAZIL

    José Barbosa Gomes

  2. ICEX, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, CP 702, 30161-970, Belo Horizonte, MG, BRAZIL

    Mário J. Dias Carneiro

  3. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, 22453-900, Rio de Janeiro, RJ, BRAZIL

    Rafael O. Ruggiero

Authors
  1. José Barbosa Gomes
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  2. Mário J. Dias Carneiro
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  3. Rafael O. Ruggiero
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Correspondence to Rafael O. Ruggiero.

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Gomes, J.B., Dias Carneiro, M.J. & Ruggiero, R.O. Hopf conjecture holds for analytic, k-basic Finsler tori without conjugate points. Bull Braz Math Soc, New Series 46, 621–644 (2015). https://doi.org/10.1007/s00574-015-0106-x

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  • DOI: https://doi.org/10.1007/s00574-015-0106-x

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