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Author: André Minor Publisher: ISBN: 9781124661544 Category : Languages : en Pages : 68
Book Description
This work will concentrate on transversal CR embeddings of one CR manifold into another both of hypersurface type and nondegenerate. In chapter 2, in lower dimensions and when the target is the sphere, it is shown that at most two such embeddings exists up to holomorphic equivalence. Lifting such an embedding to an isometry between Fefferman bundles is discussed in chapter 3. In chapter 4 it is shown that transversal chain preserving embeddings between CR hypersurfaces must preserve CR structure. Finally in chapter 5 we discuss changes of the CR second fundamental form of a CR embedding by composition with automorphisms of the source and target.
Author: André Minor Publisher: ISBN: 9781124661544 Category : Languages : en Pages : 68
Book Description
This work will concentrate on transversal CR embeddings of one CR manifold into another both of hypersurface type and nondegenerate. In chapter 2, in lower dimensions and when the target is the sphere, it is shown that at most two such embeddings exists up to holomorphic equivalence. Lifting such an embedding to an isometry between Fefferman bundles is discussed in chapter 3. In chapter 4 it is shown that transversal chain preserving embeddings between CR hypersurfaces must preserve CR structure. Finally in chapter 5 we discuss changes of the CR second fundamental form of a CR embedding by composition with automorphisms of the source and target.
Author: Son Ngoc Duong Publisher: ISBN: 9781267335265 Category : Languages : en Pages : 58
Book Description
We investigate the geometric property of transversality of holomorphic, formal or CR mappings between real-analytic, formal or smooth generic submanifolds of complex spaces of equidimension as well as of different dimensions. In Chapter 3, we shall consider the CR transversality in equidimension case. The main purpose of this chapter is to show that a holomorphic, formal or smooth CR mapping sending a real-analytic, smooth or formal generic submanifold M into such another Mʹ is CR transversal to the target, provided that the source manifold is of finite bracket type and the mapping is of generic full rank. This result and its corollary completely resolve two questions posed by Peter Ebenfelt and Linda Preiss Rothschild in a paper from 2006. We also show that under a very mild assumption on the source manifold, the generic full rank condition imposed on the mapping is also necessary for the CR transversality to hold. This result confirms a conjecture in a paper by Bernhard Lamel and Nordine Mir. In Chapter 4, we consider the transversality of mappings when the target manifold is of higher dimension. We will restrict ourself to the situation in which both manifolds M and Mʹ are hypersurfaces in Cn1 and CN1 respectively, where 1
Author: M. Salah Baouendi Publisher: Princeton University Press ISBN: 1400883962 Category : Mathematics Languages : en Pages : 418
Book Description
This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists. One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.
Author: Giuseppe Zampieri Publisher: American Mathematical Soc. ISBN: 0821844423 Category : Mathematics Languages : en Pages : 210
Book Description
Cauchy-Riemann (CR) geometry is the study of manifolds equipped with a system of CR-type equations. Compared to the early days when the purpose of CR geometry was to supply tools for the analysis of the existence and regularity of solutions to the $\bar\partial$-Neumann problem, it has rapidly acquired a life of its own and has became an important topic in differential geometry and the study of non-linear partial differential equations. A full understanding of modern CR geometryrequires knowledge of various topics such as real/complex differential and symplectic geometry, foliation theory, the geometric theory of PDE's, and microlocal analysis. Nowadays, the subject of CR geometry is very rich in results, and the amount of material required to reach competence is daunting tograduate students who wish to learn it.
Author: Sean N. Curry Publisher: American Mathematical Soc. ISBN: 1470435446 Category : CR submanifolds Languages : en Pages : 81
Book Description
The authors develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. They define a normal tractor bundle in the ambient standard tractor bundle along the submanifold and show that the orthogonal complement of this bundle is not canonically isomorphic to the standard tractor bundle of the submanifold. By determining the subtle relationship between submanifold and ambient CR density bundles the authors are able to invariantly relate these two tractor bundles, and hence to invariantly relate the normal Cartan connections of the submanifold and ambient manifold by a tractor analogue of the Gauss formula. This also leads to CR analogues of the Gauss, Codazzi, and Ricci equations. The tractor Gauss formula includes two basic invariants of a CR embedding which, along with the submanifold and ambient curvatures, capture the jet data of the structure of a CR embedding. These objects therefore form the basic building blocks for the construction of local invariants of the embedding. From this basis the authors develop a broad calculus for the construction of the invariants and invariant differential operators of CR embedded submanifolds. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naïve methods. The authors conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.
Author: Sorin Dragomir Publisher: Springer Science & Business Media ISBN: 0817644830 Category : Mathematics Languages : en Pages : 499
Book Description
Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study
Author: Howard Jacobowitz Publisher: American Mathematical Soc. ISBN: 0821815334 Category : Mathematics Languages : en Pages : 249
Book Description
The geometry and analysis of CR manifolds is the subject of this expository work, which presents all the basic results on this topic, including results from the folklore of the subject.
Author: Peter Ebenfelt Publisher: Springer Science & Business Media ISBN: 3034600097 Category : Mathematics Languages : en Pages : 353
Book Description
This volume presents the proceedings of a conference on Several Complex Variables, PDE’s, Geometry, and their interactions held in 2008 at the University of Fribourg, Switzerland, in honor of Linda Rothschild.
Author: Alexander D. Ioffe Publisher: Springer ISBN: 3319642774 Category : Mathematics Languages : en Pages : 509
Book Description
This monograph offers the first systematic account of (metric) regularity theory in variational analysis. It presents new developments alongside classical results and demonstrates the power of the theory through applications to various problems in analysis and optimization theory. The origins of metric regularity theory can be traced back to a series of fundamental ideas and results of nonlinear functional analysis and global analysis centered around problems of existence and stability of solutions of nonlinear equations. In variational analysis, regularity theory goes far beyond the classical setting and is also concerned with non-differentiable and multi-valued operators. The present volume explores all basic aspects of the theory, from the most general problems for mappings between metric spaces to those connected with fairly concrete and important classes of operators acting in Banach and finite dimensional spaces. Written by a leading expert in the field, the book covers new and powerful techniques, which have proven to be highly efficient even in classical settings, and outlines the theory’s predominantly quantitative character, leading to a variety of new and unexpected applications. Variational Analysis of Regular Mappings is aimed at graduate students and researchers in nonlinear and functional analysis, especially those working in areas close to optimization and optimal control, and will be suitable to anyone interested in applying new concepts and ideas to operations research, control engineering and numerical analysis.
Author: André Minor
Author: André Minor
Author: Ralph Abraham
Author: Son Ngoc Duong
Author: M. Salah Baouendi
Author: Giuseppe Zampieri
Author: Sean N. Curry
Author: Sorin Dragomir
Author: Howard Jacobowitz
Author: Peter Ebenfelt
Author: Alexander D. Ioffe