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Author: Sorin Dragomir
Publisher: Springer Science & Business Media
ISBN: 0817644830
Category : Mathematics
Languages : en
Pages : 499
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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: Sorin Dragomir
Publisher: Springer Science & Business Media
ISBN: 0817644830
Category : Mathematics
Languages : en
Pages : 499
Get Book
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: Giuseppe Zampieri
Publisher: American Mathematical Soc.
ISBN: 0821844423
Category : CR submanifolds
Languages : en
Pages : 210
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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: Howard Jacobowitz
Publisher: American Mathematical Soc.
ISBN: 0821815334
Category : Mathematics
Languages : en
Pages : 237
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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. The book contains a careful exposition of seminal papers by Cartan and by Chern and Moser, and also includes chapters on the geometry of chains and circles and the existence of nonrealizable CR structures. With its detailed treatment of foundational papers, the book is especially useful in that it gathers in one volume many results that were scattered throughout the literature. Directed at mathematicians and physicists seeking to understand CR structures, this self-contained exposition is also suitable as a text for a graduate course for students interested in several complex variables, differential geometry, or partial differential equations. A particular strength is an extensive chapter that prepares the reader for Cartan's approach to differential geometry. The book assumes only the usual first-year graduate courses as background.
Author: Giuseppe Zampieri
Publisher: American Mathematical Soc.
ISBN: 9781470421878
Category : Mathematics
Languages : en
Pages : 200
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Book Description
Cauchy-Riemann (CR) geometry studies manifolds equipped with a system of CR-type equations. This study has become dynamic in differential geometry and in non-linear differential equations, but many find it challenging, particularly considering the range of topics students must master (including real/complex differential and symplectic geometry) to use CR effectively. Zampieri takes graduate students through the material in remarkably gentle fashion, first covering complex variables such as Cauchy formulas in polydiscs, Levi forms and the logarithmic supermean of the Taylor radius of holomorphic functions, real structures, including Euclidean spaces, real synthetic spaces (the Frobenius-Darboux theorem), and real/complex structures such as CR manifolds and mappings, real/complex symplectic spaces, iterated commutators (Bloom-Graham normal forms) and separate real analyticity.
Author: Elisabetta Barletta
Publisher: American Mathematical Soc.
ISBN: 0821843044
Category : Cauchy-Riemann equations
Languages : en
Pages : 270
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Book Description
The authors study the relationship between foliation theory and differential geometry and analysis on Cauchy-Riemann (CR) manifolds. The main objects of study are transversally and tangentially CR foliations, Levi foliations of CR manifolds, solutions of the Yang-Mills equations, tangentially Monge-Ampere foliations, the transverse Beltrami equations, and CR orbifolds. The novelty of the authors' approach consists in the overall use of the methods of foliation theory and choice of specific applications. Examples of such applications are Rea's holomorphic extension of Levi foliations, Stanton's holomorphic degeneracy, Boas and Straube's approximately commuting vector fields method for the study of global regularity of Neumann operators and Bergman projections in multi-dimensional complex analysis in several complex variables, as well as various applications to differential geometry. Many open problems proposed in the monograph may attract the mathematical community and lead to further applications of
Author: Sean N. Curry
Publisher: American Mathematical Soc.
ISBN: 1470435446
Category : CR submanifolds
Languages : en
Pages : 81
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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: John M. Lee
Publisher: Springer
ISBN: 3319917552
Category : Mathematics
Languages : en
Pages : 437
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Book Description
This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Author: Al Boggess
Publisher: Routledge
ISBN: 1351457586
Category : Mathematics
Languages : en
Pages : 383
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Book Description
CR Manifolds and the Tangential Cauchy Riemann Complex provides an elementary introduction to CR manifolds and the tangential Cauchy-Riemann Complex and presents some of the most important recent developments in the field. The first half of the book covers the basic definitions and background material concerning CR manifolds, CR functions, the tangential Cauchy-Riemann Complex and the Levi form. The second half of the book is devoted to two significant areas of current research. The first area is the holomorphic extension of CR functions. Both the analytic disc approach and the Fourier transform approach to this problem are presented. The second area of research is the integral kernal approach to the solvability of the tangential Cauchy-Riemann Complex. CR Manifolds and the Tangential Cauchy Riemann Complex will interest students and researchers in the field of several complex variable and partial differential equations.
Author: Jeffrey Marc Lee
Publisher: American Mathematical Soc.
ISBN: 0821848151
Category : Mathematics
Languages : en
Pages : 690
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Book Description
Differential geometry began as the study of curves and surfaces using the methods of calculus. This book offers a graduate-level introduction to the tools and structures of modern differential geometry. It includes the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, and de Rham cohomology.
Author: Kai Cieliebak
Publisher: American Mathematical Soc.
ISBN: 0821885332
Category : Mathematics
Languages : en
Pages : 379
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Book Description
This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from 'Stein to Weinstein') and its applications in the complex geometric world of Stein manifolds (the road 'back').
