Advertisement

Mathematische Annalen

, Volume 332, Issue 1, pp 121–143 | Cite as

An immersion theorem for Vaisman manifolds

  • Liviu Ornea
  • Misha VerbitskyEmail author
Article

Abstract.

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering Open image in new window , with monodromy acting on Open image in new window by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on Open image in new window . We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (ℂ n ∖0)ℤ. As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.

Keywords

Manifold Complex Manifold Hopf Manifold Holomorphic Immersion Vaisman Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexandrov, B., Ivanov, St.: Weyl structures with positive Ricci tensor. Diff. Geom. Appl. 18, 343–350 (2003)Google Scholar
  2. 2.
    Baily, W.L.: On the imbedding of V-manifolds in projective spaces. Amer. J. Math. 79, 403–430 (1957)Google Scholar
  3. 3.
    Belgun, F.A.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317, 1–40 (2000)Google Scholar
  4. 4.
    Besse, A.: Einstein Manifolds, Springer-Verlag, New York (1987)Google Scholar
  5. 5.
    Boyer, C.P., Galicki, K.: 3-Sasakian Manifolds, hep-th/9810250, also published In: Surveys in differential geometry: Essays on Einstein Manifolds. M. Wang, C. LeBrun (eds.), International Press 2000, 123–184Google Scholar
  6. 6.
    Boyer, C.P., Galicki, K.: Einstein manifolds and contact geometry. Proc. Amer. Math. Soc. 129, 2419–2430 (2001)Google Scholar
  7. 7.
    Calderbank, D., Pedersen, H.: Einstein-Weyl geometry, In: Surveys in differential geometry: Essays on Einstein Manifolds, M. Wang C. LeBrun (eds.), International Press 387–423 (2000)Google Scholar
  8. 8.
    Colţoiu, M.: q-convexity. A survey, In: Complex analysis and geometry (Trento, 1995), 83–93, Pitman Res. Notes Math. Ser., 366, Longman, Harlow, 1997Google Scholar
  9. 9.
    Demailly, J.-P.: Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93–114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991Google Scholar
  10. 10.
    Dragomir, S., Ornea, L.: Locally conformal Kähler geometry, Progress in Mathematics, 155. Birkhäuser, Boston, MA, 1998Google Scholar
  11. 11.
    Gauduchon P.: La 1-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984)Google Scholar
  12. 12.
    Gauduchon, P., Ornea, L.: Locally conformally Kähler metrics on Hopf surfaces. Ann. Inst. Fourier 48, 1107–1127 (1998)Google Scholar
  13. 13.
    Gini, R., Ornea, L., Parton, M.: Locally conformal Kähler reduction, math.DG/0208208, 25 pages. To appear in J. Reine. Angew. Math.Google Scholar
  14. 14.
    Gromov, M.: Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. No. 56 (1982), 5–99 (1983)Google Scholar
  15. 15.
    Kamishima, Y.: Locally conformal Kähler manifolds with a family of constant curvature tensors. Kumamoto J. Math. 11, 19–41 (1998)Google Scholar
  16. 16.
    Kamishima, Y., Ornea, L.: Geometric flow on compact locally conformally Kahler manifolds, math.DG/0105040, 21 pages. To appear in Tohoku Math. J.Google Scholar
  17. 17.
    Madsen, A.B., Pedersen, H., Poon, Y.S., Swann, A.: Compact Einstein-Weyl manifolds with large symmetry group. Duke Math. J. 88, 407–434 (1997)Google Scholar
  18. 18.
    Morimoto, A.: On the classification of noncompact complex abelian Lie groups. Trans. Amer. Math. Soc. 123, 200–228 (1966)Google Scholar
  19. 19.
    Myers, S.B., Steenrod, N.: The group of isometries of Riemannian manifolds. Ann. Math. 40, 400–416 (1939)Google Scholar
  20. 20.
    Ornea, L.: Weyl structures on quaternionic manifolds. A state of the art, math.DG/0105041, also in: Selected Topics in Geomety and Mathematical Physics, vol. 1, 2002, 43–80, E. Barletta ed., Univ. della Basilicata (Potenza)Google Scholar
  21. 21.
    Ornea, L., Verbitsky, M.: Structure theorem for compact Vaisman manifolds. Math. Res. Let. 10 , 799–805 (2003)Google Scholar
  22. 22.
    Petit, R.: Harmonic maps and strictly pseudoconvex CR manifolds. Comm. Anal. Geom. 10, 575–610 (2002)Google Scholar
  23. 23.
    Rukimbira, P.: Chern-Hamilton’s conjecture and K-contactness. Hous. J. of Math. 21, 709–718 (1995)Google Scholar
  24. 24.
    Sasaki, S.: On differentiable manifolds with certain structures which are closely related to almost contact structure. Tohoku Math. J. 2, 459–476 (1960)Google Scholar
  25. 25.
    Tanno, S.: The standard CR-structure of the unit tangent bundle. Tohoku Math. J. 44, 535–543 (1992)Google Scholar
  26. 26.
    Tsukada, K.: Holomorphic maps of compact generalized Hopf manifolds. Geom. Dedicata 68, 61–71 (1997)Google Scholar
  27. 27.
    Vaisman, I.: Generalized Hopf manifolds. Geom. Dedicata 13, no. 3, 231–255 (1982)Google Scholar
  28. 28.
    Vaisman, I.: A survey of generalized Hopf manifolds. Rend. Sem. Mat. Torino, Special issue (1984), 205–221Google Scholar
  29. 29.
    Verbitsky, M.: Vanishing theorems for locally conformal hyperkähler manifolds, 2003, math.DG/0302219, 41 pages.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Department of MathematicsUniversity of GlasgowGlasgowScotland

Personalised recommendations