Quantum Cohomologies on Products of Cosymplectic Manifolds and Circles

  • Yong Seung Cho
  • Young Do Chai


In this paper we study the products of cosymplectic manifolds and a unit circle, which have natural symplectic structures. We have some relations on moduli spaces, Gromov–Witten invariants, and quantum cohomologies of cosymplectic manifolds and the products. As an example we examine the cosymplectic manifold \(M=S^2\times T\times S^1\) and the product \(M\times S^1\), and we also calculate their moduli spaces, Gromov–Witten invariants, and quantum cohomologies.


Cosymplectic manifold Symplectic manifold Gromov–Witten invariant Quantum cohomology 

Mathematics Subject Classification

55S15 57R15 58D15 53D15 



This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2013004848).


  1. 1.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics 203. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  2. 2.
    Blair, D.E., Goldberg, S.I.: Topology of almost contact manifolds. J. Differ. Geom. 1, 347–354 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Calabi, E., Eckmann, B.: A class of complex manifolds which are not algebraic. Ann. Math. 58, 494–500 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cho, Y.S.: Gromov–Witten invariants on products of almost contact metric manifolds, (Preprint)Google Scholar
  5. 5.
    Cho, Y.S.: Quantum type cohomologies on cosymplectic manifolds, (Preprint)Google Scholar
  6. 6.
    Cho, Y.S.: Quantum type cohomologies on contact manifolds, (Preprint)Google Scholar
  7. 7.
    Cho, Y.S.: Generating series for symmetric product spaces. Int. J. Geom. Methods Modern Phys. 9(5), 1250045 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cho, Y.S.: Quantum cohomologies of symmetric products. In. J. Geom. Methods Modern Phys. 9(1), 1250005 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cho, Y.S.: Jacobi fields in geodesic surface congruences. Int. J. Geom. Methods Modern Phys. 7(8), 1407–1412 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cho, Y.S., Hong, S.T.: Stringy Jacobi fields in Morse theory. Phys. Rev. D 75, 127902 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cho, Y.S., Hong, S.T.: Singularities in geodesic surface congruence. Phys. Rev. D 78, 067301 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cho, Y.S., Hong, S.T.: Dynamics of stringy congruence in the early universe. Phys. Rev. D 83, 104040 (2011)CrossRefGoogle Scholar
  13. 13.
    Cho, Y.S.: Hurwitz number of triple ramified covers. J. Geom. Phys. 56(4), 542–555 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cho, Y.S.: Seiberg–Witten invariants on non-symplectic 4-manifolds. Osaka J. Math. 34, 169–173 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cho, Y.S.: Finite group actions on the moduli space of self-dual connections. Trans. A.M.S. 323, 233–261 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fukaya, K., Ono, K.: Arnold conjecture and Gromov–Witten invariant. Topology 38(5), 933–1048 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Janssens, D., Vanhecke, J.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kontsevich, M., Manin, Y.: Gromov–Witten classes, quantum cohomology and enumerative geometry. Comm. Math. Phys. 164, 525–562 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    McDuff, D., Salamon, D.: J-Holomorphic Curves and Quantum Cohomology. University lecture series. A.M.S, Provindence (1994)CrossRefzbMATHGoogle Scholar
  20. 20.
    Tshikuna-Matamba, T.: Induced structures on the product of Riemannian manifolds. Int. Elect. J. Geom. 4(1), 15–25 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411–449 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2016

Authors and Affiliations

  1. 1.Division of Mathematical and Physical Sciences, College of Natural SciencesEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Department of Mathematics, College of Natural SciencesSungkyunkwan UniversitySuwonRepublic of Korea

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