On Locally Conformal Kahler Space Forms

An m-dimensional locally conformal Khler manifold (l.c.K-manifold) is characterized as a Hermitian manifold admitting a global closed 1-form a%(called the Lee form) whose structure (F%,g%) satisfies VF -8g + 8g F + aF, where ? denotes the covariant differentiation with respect to the Hermitian metric gl, 8 -Fl a, Fl F gel and the indices 9, ,l run over the range 1,2, m. For l.c.K-manifolds, I.Vaisman [4] gave a typical example and T.Kashiwada ([I], [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 gl will be expressed without the tensor field Pl. 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.