On locally conformal Kähler space forms

Source abstract

An m ‐dimensional locally conformal Kähler manifold (l.c.K‐manifold) is characterized as a Hermitian manifold admitting a global closed l‐form α λ (called the Lee form) whose structure satisfies urn:x-wiley:01611712:media:ijmm569581:ijmm569581-math-0002 , where ∇ λ denotes the covariant differentiation with respect to the Hermitian metric g μ λ , , 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 g μ λ will be expressed without the tensor field P μ λ . 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.