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Cohomological tautness of singular Riemannian foliations

  • José Ignacio Royo PrietoEmail author
  • Martintxo Saralegi-Aranguren
  • Robert Wolak
Original Paper
  • 18 Downloads

Abstract

For a Riemannian foliation \({\mathcal {F}}\) on a compact manifold M, J. A. Álvarez López proved that the geometrical tautness of \({\mathcal {F}}\), that is, the existence of a Riemannian metric making all the leaves minimal submanifolds of M, can be characterized by the vanishing of a basic cohomology class \({\varvec{\kappa }}_M\in H^1(M/{\mathcal {F}})\) (the Álvarez class). In this work we generalize this result to the case of a singular Riemannian foliation \({\mathcal {K}}\) on a compact manifold X. In the singular case, no bundle-like metric on X can make all the leaves of \({\mathcal {K}}\) minimal. In this work, we prove that the Álvarez classes of the strata can be glued in a unique global Álvarez class \({\varvec{\kappa }}_X\in H^1(X/{\mathcal {K}})\). As a corollary, if X is simply connected, then the restriction of \({\mathcal {K}}\) to each stratum is geometrically taut, thus generalizing a celebrated result of E. Ghys for the regular case.

Keywords

Singular Riemannian foliations Foliations Tautness 

Mathematics Subject Classification

53C12 57R30 37C85 

Notes

Acknowledgements

The authors wish to thank the referees for their useful remarks and suggestions.

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of the Basque Country UPV/EHUBilbaoSpain
  2. 2.Fèdèration CNRS, Nord-Pas-de-Calais FR 2956, UPRES-EA 2462 LML, Faculté Jean PerrinUniversité d’ArtoisLens CedexFrance
  3. 3.Instytut MatematykiUniwersytet JagiellonskiKrakówPoland

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