Diastases and real analytic functions on complex manifolds

Let M be a complex manifold with the complex structure J . A J-invariant symmetric tensor g is called a Kaehler tensor if the associated 2-form ωg(X, Y) =g(X, JY)(X, Y∈ TM) is closed. In addition, if g is non-degenerate it is called an indefinite Kaehler metric. A Kaehler tensor is called analytic if it is real analytic. Let Cʳ . be a complex linear space CN(N=r+s) with the indefinite Kaehler metric: gr.s=2 \∑σ=1ʳdξσ⊗ dξ̄σ-∑σ=1ˢdξσ+r⊗ dξ̄σ+r\ , where (ξ¹, ⋯ , ξN) denotes the canonical complex coordinate system. E. Calabi [1] gave a necessary and sufficient condition for a Kaehler mani- fold to be locally immersed into a complex space form as a Kaehler submanifold, and showed the rigidity of such immersions. In this paper we discuss the existence and the local rigidity of a full holomorphic mapping of M into Cr.s pre- serving a Kaehler tensor, and give several applications, where "full" means that the image of the mapping does not lie in any complex hyperplane in Cr.s .