Abstract
In this chapter we describe the diastasis function, a basic tool introduced by Calabi (Ann Math 58:1–23, 1953) which is fundamental to study Kahler immersions of Kahler manifolds into complex space forms . In Sect. 1.1 we define the diastasis function and summarize its basic properties, while in Sect. 1.2 we describe the diastasis functions of complex space forms, which represent the basic examples of Kahler manifolds. Finally, in Sect. 1.3 we give the formal definition of what a Kahler immersion is and prove that the indefinite Hilbert space constitutes a universal Kahler manifold, in the sense that it is a space in which every real analytic Kahler manifold can be locally Kahler immersed.
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