Introduction To Riemannian Manifolds
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Introduction To Riemannian Manifolds
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Author : John M. Lee
language : en
Publisher: Springer
Release Date : 2018-08-24
Introduction To Riemannian Manifolds written by John M. Lee and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-24 with Mathematics categories.
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.
Riemannian Manifolds
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Author : John M. Lee
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-06
Riemannian Manifolds written by John M. Lee and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-06 with Mathematics categories.
This book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar with topological and differentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. The author has selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all efforts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing the total curvature of a surface in term so fits topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints.
Differential And Riemannian Manifolds
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Author : Serge Lang
language : en
Publisher: Springer Science & Business Media
Release Date : 1995-03-09
Differential And Riemannian Manifolds written by Serge Lang and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-03-09 with Mathematics categories.
This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like.
Introduction To Smooth Manifolds
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Author : John M. Lee
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Introduction To Smooth Manifolds written by John M. Lee and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for under standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com puter graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics-theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible. Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the lO-dimensional manifold of 5 x 5 orthogonal ma trices, as easily as we think about the familiar 2-dimensional sphere in ]R3.
An Introduction To The Analysis Of Paths On A Riemannian Manifold
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Author : Daniel W. Stroock
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
An Introduction To The Analysis Of Paths On A Riemannian Manifold written by Daniel W. Stroock 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 2000 with Mathematics categories.
Hoping to make the text more accessible to readers not schooled in the probabalistic tradition, Stroock (affiliation unspecified) emphasizes the geometric over the stochastic analysis of differential manifolds. Chapters deconstruct Brownian paths, diffusions in Euclidean space, intrinsic and extrinsic Riemannian geometry, Bocher's identity, and the bundle of orthonormal frames. The volume humbly concludes with an "admission of defeat" in regard to recovering the Li-Yau basic differential inequality. Annotation copyrighted by Book News, Inc., Portland, OR.
An Introduction To Differentiable Manifolds And Riemannian Geometry
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Author :
language : en
Publisher: Academic Press
Release Date : 1975-08-22
An Introduction To Differentiable Manifolds And Riemannian Geometry written by and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1975-08-22 with Mathematics categories.
An Introduction to Differentiable Manifolds and Riemannian Geometry
The Laplacian On A Riemannian Manifold
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Author : Steven Rosenberg
language : en
Publisher: Cambridge University Press
Release Date : 1997-01-09
The Laplacian On A Riemannian Manifold written by Steven Rosenberg and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-09 with Mathematics categories.
This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.
Introduction To Topological Manifolds
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Author : John M. Lee
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-06
Introduction To Topological Manifolds written by John M. Lee and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-06 with Mathematics categories.
This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of di?erential geometry, algebraic topology, and related ?elds. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. Here at the University of Washington, for example, this text is used for the ?rst third of a year-long course on the geometry and topology of manifolds; the remaining two-thirds focuses on smooth manifolds. Therearemanysuperbtextsongeneralandalgebraictopologyavailable. Why add another one to the catalog? The answer lies in my particular visionofgraduateeducation—itismy(admittedlybiased)beliefthatevery serious student of mathematics needs to know manifolds intimately, in the same way that most students come to know the integers, the real numbers, Euclidean spaces, groups, rings, and ?elds. Manifolds play a role in nearly every major branch of mathematics (as I illustrate in Chapter 1), and specialists in many ?elds ?nd themselves using concepts and terminology fromtopologyandmanifoldtheoryonadailybasis. Manifoldsarethuspart of the basic vocabulary of mathematics, and need to be part of the basic graduate education. The ?rst steps must be topological, and are embodied in this book; in most cases, they should be complemented by material on smooth manifolds, vector ?elds, di?erential forms, and the like. (After all, few of the really interesting applications of manifold theory are possible without using tools from calculus.