International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 1, Pages 69-74

On locally conformal Kähler space forms

Koji Matsumoto

Department of Mathematics, Faculty of Education, Yamagata University, Yamagata 990, Japan

Received 29 May 1984

Copyright © 1985 Koji Matsumoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An m-dimensional locally conformal Kähler manifold (l.c.K-manifold) is characterized as a Hermitian manifold admitting a global closed l-form αλ (called the Lee form) whose structure (Fμλ,gμλ) satisfies νFμλ=βμgνλ+βλgνμαμFνλ+αλFνμ, where λ denotes the covariant differentiation with respect to the Hermitian metric gμλ, βλ=Fλϵαϵ, Fμλ=Fμϵgϵλ and the indices ν,μ,,λ run over the range 1,2,,m.

For l. c. K-manifolds, I. Vaisman [4] gave a typical example and T. Kashiwada ([1], [2],[3]) gave a lot of interesting properties about such manifolds.

In this paper, we shall study certain properties of l. c. K-space forms. In §2, we shall mainly get the necessary and sufficient condition that an l. c. K-space form is an Einstein one and the Riemannian curvature tensor with respect to gμλ will be expressed without the tensor field Pμλ. In §3, we shall get the necessary and sufficient condition that the length of the Lee form is constant and the sufficient condition that a compact l. c. K-space form becomes a complex space form. In the last §4, we shall prove that there does not exist a non-trivial recurrent l. c. K-space form.