Lectures On The Geometry Of Manifolds
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Lectures On The Geometry Of Manifolds
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Author : Liviu I Nicolaescu
language : en
Publisher: World Scientific
Release Date : 1996-11-13
Lectures On The Geometry Of Manifolds written by Liviu I Nicolaescu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-11-13 with Mathematics categories.
The object of this book is to introduce the reader to some of the most important techniques of modern global geometry. In writing it we had in mind the beginning graduate student willing to specialize in this very challenging field of mathematics. The necessary prerequisite is a good knowledge of the calculus with several variables, linear algebra and some elementary point-set topology.We tried to address several issues. 1. The Language; 2. The Problems; 3. The Methods; 4. The Answers.Historically, the problems came first, then came the methods and the language while the answers came last. The space constraints forced us to change this order and we had to painfully restrict our selection of topics to be covered. This process always involves a loss of intuition and we tried to balance this by offering as many examples and pictures as often as possible. We test most of our results and techniques on two basic classes examples: surfaces (which can be easily visualized) and Lie groups (which can be elegantly algebraized). When possible we present several facets of the same issue.We believe that a good familiarity with the formalism of differential geometry is absolutely necessary in understanding and solving concrete problems and this is why we presented it in some detail. Every new concept is supported by concrete examples interesting not only from an academic point of view.Our interest is mainly in global questions and in particular the interdependencegeometry ↔ topology, local ↔ global.We had to develop many algebraico-topological techniques in the special context of smooth manifolds. We spent a big portion of this book discussing the DeRham cohomology and its ramifications: Poincaré duality, intersection theory, degree theory, Thom isomorphism, characteristic classes, Gauss-Bonnet etc. We tried to calculate the cohomology groups of as many as possible concrete examples and we had to do this without relying on the powerful apparatus of homotopy theory (CW-complexes etc.). Some of the proofs are not the most direct ones but the means are sometimes more interesting than the ends. For example in computing the cohomology of complex grassmannians we returned to classical invariant theory and used some brilliant but unadvertised old ideas.In the last part of the book we discuss elliptic partial differential equations. This requires a familiarity with functional analysis. We painstakingly described the proofs of elliptic Lp and Hölder estimates (assuming some deep results of harmonic analysis) for arbitrary elliptic operators with smooth coefficients. It is not a “light meal” but the ideas are useful in a large number of instances. We present a few applications of these techniques (Hodge theory, uniformization theorem). We conclude with a close look to a very important class of elliptic operators namely the Dirac operators. We discuss their algebraic structure in some detail, Weizenböck formulæ and many concrete examples.
Lectures On The Geometry Of Manifolds 2nd Edition
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Author : Liviu I Nicolaescu
language : en
Publisher: World Scientific
Release Date : 2007-09-27
Lectures On The Geometry Of Manifolds 2nd Edition written by Liviu I Nicolaescu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-09-27 with Mathematics categories.
The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that “in learning the sciences examples are of more use than precepts”. We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a “global and analytical bias”. We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincaré duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand Hölder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.
Lectures On The Geometry Of Manifolds Third Edition
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Author : Liviu I Nicolaescu
language : en
Publisher: World Scientific
Release Date : 2020-10-08
Lectures On The Geometry Of Manifolds Third Edition written by Liviu I Nicolaescu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-08 with Mathematics categories.
The goal of this book is to introduce the reader to some of the main techniques, ideas and concepts frequently used in modern geometry. It starts from scratch and it covers basic topics such as differential and integral calculus on manifolds, connections on vector bundles and their curvatures, basic Riemannian geometry, calculus of variations, DeRham cohomology, integral geometry (tube and Crofton formulas), characteristic classes, elliptic equations on manifolds and Dirac operators. The new edition contains a new chapter on spectral geometry presenting recent results which appear here for the first time in printed form.
Lectures On Differential Geometry
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Author : Shiing-Shen Chern
language : en
Publisher: World Scientific
Release Date : 1999
Lectures On Differential Geometry written by Shiing-Shen Chern and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Mathematics categories.
This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives.
Lectures On Symplectic Geometry
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Author : Ana Cannas da Silva
language : en
Publisher: Springer
Release Date : 2004-10-27
Lectures On Symplectic Geometry written by Ana Cannas da Silva and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-10-27 with Mathematics categories.
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
Lectures On Differential Geometry
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Author : Weihuan Chen
language : en
Publisher: World Scientific Publishing Company
Release Date : 1999-11-30
Lectures On Differential Geometry written by Weihuan Chen and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-11-30 with Mathematics categories.
This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives.
Lectures On Differential Geometry
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Author : Shlomo Sternberg
language : en
Publisher: American Mathematical Soc.
Release Date : 1999
Lectures On Differential Geometry written by Shlomo Sternberg and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Mathematics categories.
This book is based on lectures given at Harvard University during the academic year 1960-1961. The presentation assumes knowledge of the elements of modern algebra (groups, vector spaces, etc.) and point-set topology and some elementary analysis. Rather than giving all the basic information or touching upon every topic in the field, this work treats various selected topics in differential geometry. The author concisely addresses standard material and spreads exercises throughout the text. His reprint has two additions to the original volume: a paper written jointly with V. Guillemin at the beginning of a period of intense interest in the equivalence problem and a short description from the author on results in the field that occurred between the first and the second printings.
Lectures On The Geometry Of Poisson Manifolds
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Author : Izu Vaisman
language : en
Publisher:
Release Date : 1992
Lectures On The Geometry Of Poisson Manifolds written by Izu Vaisman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with categories.
Lectures On The Geometry Of Position
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Author : Theodor Reye
language : en
Publisher:
Release Date : 1898
Lectures On The Geometry Of Position written by Theodor Reye and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1898 with Geometry categories.
Lectures On The Geometry Of Quantization
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Author : Sean Bates
language : en
Publisher: American Mathematical Soc.
Release Date : 1997
Lectures On The Geometry Of Quantization written by Sean Bates and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.
These notes are based on a course entitled ``Symplectic Geometry and Geometric Quantization'' taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading.