International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 1, Pages 69-74
doi:10.1155/S0161171285000060

On locally conformal Kähler space forms

Abstract

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+βλgνμ−αμFνλ+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.