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Core reduction for singular Riemannian foliations and applications to positive curvature

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Abstract

We expand upon the notion of a pre-section for a singular Riemannian foliation \((M,\mathcal {F})\), i.e. a proper submanifold \(N\subset M\) retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of \((M,\mathcal {F})\) and the leaf space \(M/\mathcal {F}\). In particular, we show that if a foliated manifold M has positive sectional curvature and contains a non-trivial pre-section, then the leaf space \(M/\mathcal {F}\) has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions.

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Acknowledgements

We would like to thank Fernando Galaz-Garcia, Karsten Grove, Alexander Lytchak and Marco Radeschi for helpful conversations. We thank Ricardo Mendes for useful observations on the manuscript. The second author would like to thank the Department of Mathematics at KIT for their hospitality during the visit where a portion of this work was done.

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Author notes

  1. Diego Corro

    Present address: Fakultät fur Mathematik, Karlsruher Institut fur Technologie, Englerstr. 2, 76181, Karlsruhe, Deutschland

  2. Adam Moreno

    Present address: Mathematics and Statistics Department, Amherst College, 220 South Pleasant Street, Amherst, MA, 01002, USA

Authors and Affiliations

  1. Instituto de Matemáticas, Unidad Oaxaca, Universidad Nacional Autónoma de México UNAM, Antonio de León #2, altos, Col. Centro, 68000, Oaxaca, Oaxaca de Juárez, CP, México

    Diego Corro

  2. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095-1555, USA

    Adam Moreno

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  1. Diego Corro
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  2. Adam Moreno
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Correspondence to Adam Moreno.

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Supported by the DFG (281869850, RTG 2229 “Asymptotic Invariants and Limits of Groups and Spaces”). Supported by DFG-Eigenestelle Fellowship CO 2359/1-1. Supported by a DGAPA postdoctoral Scholarship of the Institute of Mathematics - UNAM.

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Corro, D., Moreno, A. Core reduction for singular Riemannian foliations and applications to positive curvature. Ann Glob Anal Geom 62, 617–634 (2022). https://doi.org/10.1007/s10455-022-09856-y

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