CR Submanifolds of Complex Projective Space

This text begins with an introduction to complex differential geometry and the properties of complex manifolds. It then describes the properties of hypersurfaces of various complex spaces and CR manifolds, emphasizing CR submanifolds of maximal CR dimension.

Althoughsubmanifoldscomplexmanifoldshasbeenanactive?eldofstudyfor many years, in some sense this area is not su?ciently covered in the current literature. This text deals with the CR submanifolds of complex manifolds, with particular emphasis on CR submanifolds of complex projective space, and it covers the topics which are necessary for learning the basic properties of these manifolds. We are aware that it is impossible to give a complete overview of these submanifolds, but we hope that these notes can serve as an introduction to their study. We present the fundamental de?nitions and results necessary for reaching the frontiers of research in this ?eld. There are many monographs dealing with some current interesting topics in di?erential geometry, but most of these are written as encyclopedias, or research monographs, gathering recent results and giving the readers ample usefulinformationaboutthetopics. Therefore, thesekindsofmonographsare attractive to specialists in di?erential geometry and related ?elds and acce- able to professional di?erential geometers. However, for graduate students who are less advanced in di?erential geometry, these texts might be hard to read without assistance from their instructors. By contrast, the general philosophy of this book is to begin with the elementary facts about complex manifolds and their submanifolds, give some details and proofs, and introduce the reader to the study of CR submanifolds of complex manifolds; especially complex projective space. It includes only a few original results with precise proofs, while the others are cited in the reference list.

Presents many recent developments and results in the study of CR submanifolds not previously published Provides a self-contained introduction to complex differential geometry Provides relevant techniques, results, application, and insight into the motivations and ideas behind the theory Includes supplementary material: sn.pub/extras

Autorentext

Mirjana Djoric and Masafumi Okumura are widely published in the field of differential geometry. They have each contributed chapters Springer publictations and have co-published 5 papers on the topic of CR submanifolds in Springer Journals.

Klappentext

This book covers the necessary topics for learning the basic properties of complex manifolds and their submanifolds, offering an easy, friendly, and accessible introduction into the subject while aptly guiding the reader to topics of current research and to more advanced publications.

The book begins with an introduction to the geometry of complex manifolds and their submanifolds and describes the properties of hypersurfaces and CR submanifolds, with particular emphasis on CR submanifolds of maximal CR dimension. The second part contains results which are not new, but recently published in some mathematical journals. The final part contains several original results by the authors, with complete proofs.

Key features of "CR Submanifolds of Complex Projective Space":

• Presents recent developments and results in the study of submanifolds previously published only in research papers.

• Special topics explored include: the Kähler manifold, submersion and immersion, codimension reduction of a submanifold, tubes over submanifolds, geometry of hypersurfaces and CR submanifolds of maximal CR dimension.

• Provides relevant techniques, results and their applications, and presents insight into the motivations and ideas behind the theory.

• Presents the fundamental definitions and results necessary for reaching the frontiers of research in this field.

  This text is largely self-contained. Prerequisites include basic knowledge of introductory manifold theory and of curvature properties of Riemannian geometry. Advanced undergraduates, graduate students and researchers in differential geometry will benefit from this concise approach to an important topic.

  Inhalt
  Complex manifolds.- Almost complex structure.- Complex vector spaces, complexification.- K#x00E4;hler manifolds.- Structure equations of a submanifold.- Submanifolds of a Euclidean space.- Submanifolds of a complex manifold.- The Levi form.- The principal circle bundle S(P(C), S).- Submersion and immersion.- Hypersurfaces of a Riemannian manifold of constant curvature.- Hypersurfaces of a sphere.- Hypersurfaces of a sphere with parallel shape operator.- Codimension reduction of a submanifold.- CR submanifolds of maximal CR dimension.- Real hypersurfaces of a complex projective space.- Tubes over submanifolds.- Levi form of CR submanifolds of maximal CR dimension of a complex space form.- Eigenvalues of the shape operator of CR submanifolds of maximal CR dimension of a complex space form.- CR submanifolds of maximal CR dimension satisfying the condition (, ) + (, ) = 0.- Contact CR submanifolds of maximal CR dimension.- Invariant submanifolds of real hypersurfaces of complex space forms.- The scalar curvature of CR submanifolds of maximal CR dimension.