ON LOCALLY CONFORMAL KAHLER SPACE FORMS

An m-dimensional locally conformal Kaler manifold (l. c. K-manifold) is characterized as a Hermitian manifold admitting a global closed 1-form α_λ(called the Lee form) whose structure (Fμ_^λ,g_(μλ)) satisfies ∇_νF_(μλ)=-β_μg_(νλ) +β_λg_(νμ) -α_μF_(νλ) +α_λF_(νμ), where ∇_λdenotes the covariant differentiation with respect to the Hermitian metric g_(μλ), β_λ=-F_λ^ε α_ε, F_(μλ) = F_μ^ε g_(ελ) and the indicesν,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_(μλ) will be expressed without the tensor field P_(μλ). 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.

並列關鍵字

L.c.K-manifolds ； Lee form ； l.c.K-space forms ； hybrid ； recurrent I. c. K-space form

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