New examples of compact cosymplectic solvmanifolds

J. C. Marrero; E. Padrón-Fernández

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 3, page 337-345
  • ISSN: 0044-8753

Abstract

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In this paper we present new examples of ( 2 n + 1 ) -dimensional compact cosymplectic manifolds which are not topologically equivalent to the canonical examples, i.e., to the product of the ( 2 m + 1 ) -dimensional real torus and the r -dimensional complex projective space, with m , r ≥ 0 and m + r = n . These new examples are compact solvmanifolds and they are constructed as suspensions with fibre the 2 n -dimensional real torus. In the particular case n = 1 , using the examples obtained, we conclude that a 3 -dimensional compact flat orientable Riemannian manifold with non-zero first Betti number admits a cosymplectic structure. Furthermore, if the first Betti number is equal to 1 then such a manifold is not topologically equivalent to the global product of a compact Kähler manifold with the circle S 1 .

How to cite

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Marrero, J. C., and Padrón-Fernández, E.. "New examples of compact cosymplectic solvmanifolds." Archivum Mathematicum 034.3 (1998): 337-345. <http://eudml.org/doc/248176>.

@article{Marrero1998,
abstract = {In this paper we present new examples of $(2n+1)$-dimensional compact cosymplectic manifolds which are not topologically equivalent to the canonical examples, i.e., to the product of the $(2m+1)$-dimensional real torus and the $r$-dimensional complex projective space, with $m,r\ge 0$ and $m+r=n.$ These new examples are compact solvmanifolds and they are constructed as suspensions with fibre the $2n$-dimensional real torus. In the particular case $n=1,$ using the examples obtained, we conclude that a $3$-dimensional compact flat orientable Riemannian manifold with non-zero first Betti number admits a cosymplectic structure. Furthermore, if the first Betti number is equal to $1$ then such a manifold is not topologically equivalent to the global product of a compact Kähler manifold with the circle $S^1.$},
author = {Marrero, J. C., Padrón-Fernández, E.},
journal = {Archivum Mathematicum},
keywords = {cosymplectic manifolds; solvmanifolds; Kähler manifolds; suspensions; flat Riemannian manifolds; cosymplectic manifold; solvmanifold; Kähler manifold; suspension; flat Riemannian manifold},
language = {eng},
number = {3},
pages = {337-345},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {New examples of compact cosymplectic solvmanifolds},
url = {http://eudml.org/doc/248176},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Marrero, J. C.
AU - Padrón-Fernández, E.
TI - New examples of compact cosymplectic solvmanifolds
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 3
SP - 337
EP - 345
AB - In this paper we present new examples of $(2n+1)$-dimensional compact cosymplectic manifolds which are not topologically equivalent to the canonical examples, i.e., to the product of the $(2m+1)$-dimensional real torus and the $r$-dimensional complex projective space, with $m,r\ge 0$ and $m+r=n.$ These new examples are compact solvmanifolds and they are constructed as suspensions with fibre the $2n$-dimensional real torus. In the particular case $n=1,$ using the examples obtained, we conclude that a $3$-dimensional compact flat orientable Riemannian manifold with non-zero first Betti number admits a cosymplectic structure. Furthermore, if the first Betti number is equal to $1$ then such a manifold is not topologically equivalent to the global product of a compact Kähler manifold with the circle $S^1.$
LA - eng
KW - cosymplectic manifolds; solvmanifolds; Kähler manifolds; suspensions; flat Riemannian manifolds; cosymplectic manifold; solvmanifold; Kähler manifold; suspension; flat Riemannian manifold
UR - http://eudml.org/doc/248176
ER -

References

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  1. Blair D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, Berlin, (1976). (1976) Zbl0319.53026MR0467588
  2. Blair D. E., Goldberg S. I., Topology of almost contact manifolds, J. Diff. Geometry, 1, 347-354 (1967). (1967) Zbl0163.43902MR0226539
  3. Chinea D., León M. de, Marrero J. C., Topology of cosymplectic manifolds, J. Math. Pures Appl., 72, 567-591 (1993). (1993) Zbl0845.53025MR1249410
  4. Hector G., Hirsch U., Introduction to the Geometry of Foliations. Part A, Aspects of Math., Friedr. Vieweg and Sohn, (1981). (1981) Zbl0486.57002MR0639738
  5. León M. de, Marrero J. C., Compact cosymplectic manifolds with transversally positive definite Ricci tensor, Rendiconti di Matematica, Serie VII, 17 Roma, 607-624 (1997). (1997) Zbl0897.53026MR1620868
  6. Wolf J. A., Spaces of constant curvature, 5nd ed., Publish or Perish, Inc., Wilmington, Delaware, (1984). (1984) Zbl0556.53033MR0928600

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