# Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds

Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds A compact complex manifold V is called Vaisman if it admits a Hermitian metric which is conformal to a Kähler one, and a non-isometric conformal action by C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}$$\end{document}. It is called quasi-regular if the C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}$$\end{document}-action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of V. It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as a quasi-regular quotient of V. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold M is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as an S1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S^1$$\end{document}-quotient of M. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

# Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds

, Volume OnlineFirst – May 17, 2021
10 pages

/lp/springer-journals/closed-orbits-of-reeb-fields-on-sasakian-manifolds-and-elliptic-curves-ZcvAPBU2d3
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-021-02776-w
Publisher site
See Article on Publisher Site

### Abstract

A compact complex manifold V is called Vaisman if it admits a Hermitian metric which is conformal to a Kähler one, and a non-isometric conformal action by C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}$$\end{document}. It is called quasi-regular if the C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}$$\end{document}-action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of V. It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as a quasi-regular quotient of V. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold M is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as an S1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S^1$$\end{document}-quotient of M.

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 17, 2021