Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 37–55 | Cite as

Isometric immersions of locally conformally Kähler manifolds

  • Daniele Angella
  • Michela ZeddaEmail author


We investigate isometric immersions of locally conformally Kähler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.


Locally conformally Kähler Isometric immersion Calabi’s diastasis function Hopf manifold Compact complex surface 

Mathematics Subject Classification

53B35 53C55 32H02 53A30 



The authors are very grateful to Andrea Loi and Liviu Ornea for the interest in their work and for all the stimulating conversations. We would also like to thank the anonymous referees for having pointed out some mistakes in a previous version and for their very useful suggestions.


  1. 1.
    Bazzoni, G.: Vaisman nilmanifolds. Bull. Lond. Math. Soc. 49(5), 824–830 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Belgun, F.A.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317(1), 1–40 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brunella, M.: Locally conformally Kähler metrics on Kato surfaces. Nagoya Math. J. 202, 77–81 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calabi, E.: Isometric imbedding of complex manifolds. Ann. Math. (2) 58, 1–23 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cordero, L.A., Fernández, M., de León, M.: Compact locally conformal Kähler nilmanifolds. Geom. Dedicata 21(2), 187–192 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Di Scala, A.J., Ishi, H., Loi, A.: Kähler immersions of homogeneous Kähler manifolds into complex space forms. Asian J. Math. 16(3), 479–487 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dragomir, S., Ornea, L.: Locally Conformal Kähler Geometry, Progress in Mathematics, vol. 155. Birkhäuser, Boston (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gauduchon, P., Ornea, L.: Locally conformally Kähler metrics on Hopf surfaces. Ann. Inst. Fourier (Grenoble) 48(4), 1107–1127 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Inoue, M.: On surfaces of class \(VII_{0}\). Invent. Math. 24, 269–310 (1974)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kodaira, K.: On the structure of compact complex analytic surfaces. I. Amer. J. Math. 86, 751–798 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kodaira, K.: On the structure of compact complex analytic surfaces. II. Amer. J. Math. 88, 682–721 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kodaira, K.: On the structure of compact complex analytic surfaces. III. Amer. J. Math. 90, 55–83 (1968)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kodaira, K.: On the structure of compact complex analytic surfaces. IV. Amer. J. Math. 90, 1048–1066 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Loi, A., Zedda, M.: Kähler immersions of Kähler manifolds into complex space forms. arXiv:1712.04298 [math.DG]
  15. 15.
    Ornea, L.: Locally conformally Kähler manifolds. A selection of results, Lecture notes of Seminario Interdisciplinare di Matematica, Vol. IV, pp. 121–152, Lecture Notes in Seminor Interdisciplinary Matematics, S.I.M., Potenza (2005)Google Scholar
  16. 16.
    Ornea, L., Verbitsky, M.: An immersion theorem for Vaisman manifolds. Math. Ann. 332(1), 121–143 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ornea, L., Verbitsky, M.: Morse-Novikov cohomology of locally conformally Kähler manifolds. J. Geom. Phys. 59(3), 295–305 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ornea, L., Verbitsky, M.: Locally conformal Kähler manifolds with potential. Math. Ann. 348, 25–33 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ornea, L., Verbitsky, M.: Topology of locally conformally Kähler manifolds with potential. Int. Math. Res. Not. IMRN 2010(4), 717–726 (2010)zbMATHGoogle Scholar
  20. 20.
    Ornea, L., Verbitsky, M.: LCK rank of locally conformally Kähler manifolds with potential. J. Geom. Phys. 107, 92–98 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ornea, L., Verbitsky, M.: Embedding of LCK manifolds with potential into Hopf manifolds using Riesz–Schauder theorem. In Complex and Symplectic Geometry, pp. 137–148, Springer INdAM Series, vol. 21. Springer, Berlin (2017)Google Scholar
  22. 22.
    Parton, M.: Hopf surfaces: locally conformal Kähler metrics and foliations. Ann. Mat. Pura Appl. (4) 182(3), 287–306 (2003). arXiv version at arXiv:math/9907105v1 [math.DG]
  23. 23.
    Parton, M., Vuletescu, V.: Examples of non-trivial rank in locally conformal Kähler geometry. Math. Z. 270(1–2), 179–187 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pontecorvo, M.: On a question of Vaisman concerning complex surfaces. Ann. Mat. Pura Appl. (4) 193(5), 1283–1293 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ruan, W.-D.: Canonical coordinates and Bergmann metrics. Comm. Anal. Geom. 6(3), 589–631 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32(1), 99–130 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tricerri, F.: Some examples of locally conformal Kähler manifolds. Rend. Semin. Mat. Univ. Politec. Torino 40(1), 81–92 (1982)zbMATHGoogle Scholar
  28. 28.
    Ugarte, L.: Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vaisman, I.: Non-Kähler metrics on geometric complex surfaces. Rend. Semin. Mat. Univ. Politec. Torino 45(3), 117–123 (1987)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Verbitsky, M.: Vanishing theorems for locally conformal hyperkaehler manifolds. Tr. Mat. Inst. Steklova 246 (2004). In Algebr. Geom. Metody, Svyazi i Prilozh, pp. 64–91 (transl: Proc. Steklov Inst. Math. 246(3), 54–78, 2004)Google Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “Ulisse Dini”Università degli Studi di FirenzeFlorenceItaly
  2. 2.Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Plesso Matematico e InformaticoUniversità di ParmaParmaItaly

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