The aim of this book is to describe Calabi's original work on Kähler immersions of Kähler
manifolds into complex space forms to provide a detailed account of what is known today on the
subject and to point out some open problems. Calabi's pioneering work making use of the
powerful tool of the diastasis function allowed him to obtain necessary and sufficient
conditions for a neighbourhood of a point to be locally Kähler immersed into a finite or
infinite-dimensional complex space form. This led to a classification of (finite-dimensional)
complex space forms admitting a Kähler immersion into another and to decades of further
research on the subject. Each chapter begins with a brief summary of the topics to be discussed
and ends with a list of exercises designed to test the reader's understanding. Apart from the
section on Kähler immersions of homogeneous bounded domains into the infinite complex
projective space which could be skipped without compromising the understanding of the rest of
the book the prerequisites to read this book are a basic knowledge of complex and Kähler
geometry.
