Vertical–horizontal decomposition of Laplacians and cohomologies of manifolds with trivial tangent bundles

  • Sönke Rollenske
  • Adriano TomassiniEmail author
  • Xu Wang


In this paper, we obtain a vertical–horizontal decomposition formula of Laplacians on manifolds with a special foliation structure. Two Nomizu-type theorems for cohomologies of nilmanifolds follow as applications.


Künneth formula Horizontal lift Vertical form Laplacian Nilpotent foliation Torus fibration Nomizu-type theorem Nilpotent group 

Mathematics Subject Classification

53C25 53C55 



Xu Wang would like to thank B. Berndtsson, J.P. Demailly and N. Mok for several useful discussions about the topics of this paper. Sönke Rollenske is grateful to the other authors for the invitation to join the project at a relatively late stage. He is also grateful to A. Fino and J. Ruppenthal for many discussions about the Dolbeault cohomology of nilmanifolds that culminated in the paper [13]. We are also pleased to thank the anonymous referee for valuable remarks and suggestions for a better presentation of our results.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.FB 12/Mathematik und InformatikPhilipps–Universität MarburgMarburgGermany
  2. 2.Dipartimento di Scienze Matematiche, Fisiche e InformaticheUnità di Matematica e Informatica, Universitá degli Studi di ParmaParmaItaly
  3. 3.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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