Differential Geometry and Analysis on CR Manifolds


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




Complex Analysis and CR Geometry


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.




Introduction to Riemannian Manifolds


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.




From Stein to Weinstein and Back


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').




On the Hypotheses Which Lie at the Bases of Geometry


Book Description

This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.







An Introduction to Manifolds


Book Description

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.




Geometry of Manifolds


Book Description

From the Preface of the First Edition: ``Our purpose in writing this book is to put material which we found stimulating and interesting as graduate students into form. It is intended for individual study and for use as a text for graduate level courses such as the one from which this material stems, given by Professor W. Ambrose at MIT in 1958-1959. Previously the material had been organized in roughly the same form by him and Professor I. M. Singer, and they in turn drew upon thework of Ehresmann, Chern, and E. Cartan. Our contributions have been primarily to fill out the material with details, asides and problems, and to alter notation slightly. ``We believe that this subject matter, besides being an interesting area for specialization, lends itself especially to a synthesisof several branches of mathematics, and thus should be studied by a wide spectrum of graduate students so as to break away from narrow specialization and see how their own fields are related and applied in other fields. We feel that at least part of this subject should be of interest not only to those working in geometry, but also to those in analysis, topology, algebra, and even probability and astronomy. In order that this book be meaningful, the reader's background should include realvariable theory, linear algebra, and point set topology.'' This volume is a reprint with few corrections of the original work published in 1964. Starting with the notion of differential manifolds, the first six chapters lay a foundation for the study of Riemannian manifolds through specializing the theoryof connections on principle bundles and affine connections. The geometry of Riemannian manifolds is emphasized, as opposed to global analysis, so that the theorems of Hopf-Rinow, Hadamard-Cartan, and Cartan's local isometry theorem are included, but no elliptic operator theory. Isometric immersions are treated elegantly and from a global viewpoint. In the final chapter are the more complicated estimates on which much of the research in Riemannian geometry is based: the Morse index theorem,Synge's theorems on closed geodesics, Rauch's comparison theorem, and the original proof of the Bishop volume-comparison theorem (with Myer's Theorem as a corollary). The first edition of this book was the origin of a modern treatment of global Riemannian geometry, using the carefully conceived notationthat has withstood the test of time. The primary source material for the book were the papers and course notes of brilliant geometers, including E. Cartan, C. Ehresmann, I. M. Singer, and W. Ambrose. It is tightly organized, uniformly very precise, and amazingly comprehensive for its length.




Differential Geometry and Statistics


Book Description

Ever since the introduction by Rao in 1945 of the Fisher information metric on a family of probability distributions, there has been interest among statisticians in the application of differential geometry to statistics. This interest has increased rapidly in the last couple of decades with the work of a large number of researchers. Until now an impediment to the spread of these ideas into the wider community of statisticians has been the lack of a suitable text introducing the modern coordinate free approach to differential geometry in a manner accessible to statisticians. Differential Geometry and Statistics aims to fill this gap. The authors bring to this book extensive research experience in differential geometry and its application to statistics. The book commences with the study of the simplest differentiable manifolds - affine spaces and their relevance to exponential families, and goes on to the general theory, the Fisher information metric, the Amari connections and asymptotics. It culminates in the theory of vector bundles, principal bundles and jets and their applications to the theory of strings - a topic presently at the cutting edge of research in statistics and differential geometry.




Locally Conformal Kähler Geometry


Book Description

. E C, 0 1'1 1, and n E Z, n ~ 2. Let~.. be the O-dimensional Lie n group generated by the transformation z ~ >.z, z E C - {a}. Then (cf.