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Transactions of the American Mathematical Society

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

 
 

 

Positively curved Killing foliations via deformations


Authors: Francisco C. Caramello Jr. and Dirk Töben
Journal: Trans. Amer. Math. Soc. 372 (2019), 8131-8158
MSC (2010): Primary 53C12; Secondary 57R30
DOI: https://doi.org/10.1090/tran/7893
Published electronically: July 30, 2019
MathSciNet review: 4029693
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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

Keywords: Riemannian foliations, Killing foliations, deformations, positive transverse curvature, transverse symmetry rank, basic Euler characteristic
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.
Article copyright: © Copyright 2019 American Mathematical Society