Kähler Immersions of Kähler Manifolds into Complex Space Forms

Abstract

The study of Kähler immersions of a given real analytic Kähler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows 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. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a Kähler immersion into another. Although, a complete classification of Kähler manifolds admitting a Kähler immersion into complex space forms is not known, not even when the Kähler manifolds involved are of great interest, e.g. when they are Kähler-Einstein or homogeneous spaces.In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application.Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained.The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.Each chapter begins with a brief summary of the topics discussed and ends with a list of exercises which help the reader to test his understanding.Apart from the topics discussed in Section 3.1 of Chapter 3, which could be skipped without compromising the understanding of the rest of the book, the requirements to read this book are a basic knowledge of complex and Kähler iii geometry (treated, e.g. in Moroianu's book [59]).The authors are grateful to Claudio Arezzo and Fabio Zuddas for a careful reading of the text and for valuable comments that have improved the book's exposure.

Diastatic entropy and rigidity of complex hyperbolic manifolds

“…Let ϕ : (Y, g) → (N, g N ) be the strongly proper Kähler immersion of Y in an locally classical symmetric space of noncompact type N and let ϕ : ( Y , g) → (Ω, g Ω ) be its lift to the Kähler universal covers. By (9) we see that ( Y , g) has the diastasis globally defined. As lim t→+∞…”

“…Moreover, when we fix one of its entries, let's say p, then the diastasis centred in p, D p : U → R given by D p (q) := D (p, q) is a Kähler potential. The reader is referred to [9] for further details and for an updated account on projectively induced Kähler metrics.…”

η-Einstein Sasakian immersions in non-compact Sasakian space forms

“…For the second part of the theorem, let us suppose U p = S. Reasoning as before, by completeness of K and by Calabi 's Rigidity Theorem [6] for Kähler immersions into Kähler space forms (see also [19]) one obtains the stronger result that either K = C n or K = B n and the projection is just the trivial fibration because in both cases K contractible. Then, since S is complete, the fibres of the fibration are diffeomorphic either to R or to S 1 .…”

Einstein and $$\eta $$-Einstein Sasakian submanifolds in spheres

“…This definition was first considered in the early seventies, under different names, by Okumura [30], Harada [15][16][17], Kon [23,24], who mainly studied some geometric conditions ensuring the immersed manifold to be totally geodesic. However, despite the theory of Kähler immersions, which has widely developed in the last decades due to the fundamental work of Calabi (see [19] for an updated review of this topic), there are very few results about Sasakian immersions. Relapsing some conditions in (8)-( 9), we can mention a recent, remarkable result of Ornea and Verbitsky [29].…”