Annals of Global Analysis and Geometry

, Volume 13, Issue 1, pp 43–54 | Cite as

Locally conformally Hermitian-flat manifolds

  • Koji Matsuo


We introduce the notion of a locally conformally Hermitian-flat manifold and derive a necessary and sufficient condition for a Hermitian manifold to be locally conformally Hermitian-flat. In addition, we construct a family of examples.

Key words

Hermitian connection conformal invariance 

MSC 1991

53C55 53C25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Balas, A.: Compact Hermitian manifolds of constant holomorphic sectional curvature.Math. Z. 189 (1985), 193–210.Google Scholar
  2. [2]
    Balas, A.;Gauduchon, P.: Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler.Math. Z. 190 (1985), 39–43.Google Scholar
  3. [3]
    Besse, A.L.:Einstein Manifolds. Ergebnisse der Math., 3. Folge, Bd. 10, Springer-Verlag, Berlin, Heidelberg, New York 1987.Google Scholar
  4. [4]
    Boothby, W.M.: Hermitian manifolds with zero curvature.Mich. Math. J. 5 (1958), 229–233.Google Scholar
  5. [5]
    Gauduchon, P.: La 1-forme de torsion d'une variété hermitienne compacte.Math. Ann. 267 (1984), 495–518.Google Scholar
  6. [6]
    Goldberg, S.I.:Curvature and Homology. Academic Press, New York 1962.Google Scholar
  7. [7]
    Kitahara, H.;Matsuo, K.;Pak, J.S.: A conformal curvature tensor field on Hermitian manifolds.J. Korean Math. Soc. 27 (1990), 7–17.Google Scholar
  8. [8]
    Kitahara, H.;Matsuo, K.;Pak, J.S.: Appendium: A conformal curvature tensor field on Hermitian manifolds.Bull. Korean Math. Soc. 27 (1990), 27–30.Google Scholar
  9. [9]
    Kobayashi, S.;Nomizu, K.:Foundations of Differential Geometry. Interscience Publishers, New York 1969.Google Scholar
  10. [10]
    Kuiper, N.H.: On conformally flat spaces in the large.Ann. Math. 50 (1949), 916–924.Google Scholar
  11. [11]
    Tanno, S.: An inequality for 4-dimensional Kählerian manifolds.Proc. Japan Acad. 49 (1973), 257–261.Google Scholar
  12. [12]
    Vaisman, I.: On locally conformal almost Kähler manifolds.Isr. J. Math. 24 (1976), 338–351.Google Scholar
  13. [13]
    Vaisman, I.: A theorem on compact locally conformal Kähler manifolds.Proc. Amer. Math. Soc. 75 (1979), 279–283.Google Scholar
  14. [14]
    Vaisman, I.: On locally and globally conformal Kähler manifolds.Trans. Am. Math. Soc. 262 (1980), 533–542.Google Scholar
  15. [15]
    Vaisman, I.: Generalized Hopf manifolds.Geom. Dedicata 13 (1982), 231–255.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Koji Matsuo
    • 1
  1. 1.Department of MathematicsIchinoseki National College of TechnologyIchinosekiJapan

Personalised recommendations