The Diastasis Function

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 Kähler immersions of Kähler 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 Kähler manifolds. Finally, in Sect. 1.3 we give the formal definition of what a Kähler immersion is and prove that the indefinite Hilbert space constitutes a universal Kähler manifold, in the sense that it is a space in which every real analytic Kähler manifold can be locally Kähler immersed.