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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Positively curved Killing foliations via deformations
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by Francisco C. Caramello Jr. and Dirk Töben PDF
Trans. Amer. Math. Soc. 372 (2019), 8131-8158 Request permission

Abstract:

We show that a compact manifold that admits a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact, we prove that a Riemannian foliation of a compact manifold with finite fundamental group and nonvanishing Euler characteristic is closed. As another application, we obtain that, for a positively curved Killing foliation of a compact manifold, if the structural algebra has sufficiently large dimension, then the basic Euler characteristic is positive.
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Additional Information
  • Francisco C. Caramello Jr.
  • Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, Rodovia Washington Luís, Km. 235, 13565-905 São Carlos, São Paulo, Brazil
  • Email: franciscocaramello@dm.ufscar.br
  • Dirk Töben
  • Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, Rodovia Washington Luís, Km. 235, 13565-905 São Carlos, São Paulo, Brazil
  • Email: dirktoben@dm.ufscar.br
  • Received by editor(s): March 18, 2018
  • Received by editor(s) in revised form: June 8, 2018, and March 19, 2019
  • Published electronically: July 30, 2019
  • Additional Notes: The first author was supported by the Brazilian Federal Agency for Support and Evaluation of Graduate Education.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8131-8158
  • MSC (2010): Primary 53C12; Secondary 57R30
  • DOI: https://doi.org/10.1090/tran/7893
  • MathSciNet review: 4029693