Differential Geometry

Bundles, Connections, Metrics and Curvature
Author: Clifford Henry Taubes
Publisher: OUP Oxford
ISBN: 0191621226
Category: Mathematics
Page: 312
View: 9919

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Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.

An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings

Author: Frederick W. Gehring,Gaven J. Martin,Bruce P. Palka
Publisher: American Mathematical Soc.
ISBN: 0821843605
Category: Conformal mapping
Page: 116
View: 5947

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This book offers a modern, up-to-date introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, Kleinian groups, geometric topology, differential geometry, and geometric group theory. It is a careful and detailed introduction to the higher-dimensional theory of quasiconformal mappings from the geometric viewpoint, based primarily on the technique of the conformal modulus of a curve family. Notably, the final chapter describes the application of quasiconformal mapping theory to Mostow's celebrated rigidity theorem in its original context with all the necessary background. This book will be suitable as a textbook for graduate students and researchers interested in beginning to work on mapping theory problems or learning the basics of the geometric approach to quasiconformal mappings. Only a basic background in multidimensional real analysis is assumed.

Differential Geometry

Connections, Curvature, and Characteristic Classes
Author: Loring W. Tu
Publisher: Springer
ISBN: 3319550845
Category: Mathematics
Page: 347
View: 5974

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This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

Modeling Differential Equations in Biology

Author: Clifford Henry Taubes
Publisher: Cambridge University Press
ISBN: 1316582787
Category: Mathematics
Page: N.A
View: 4455

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Based on a very successful one-semester course taught at Harvard, this text teaches students in the life sciences how to use differential equations to help their research. It needs only a semester's background in calculus. Ideas from linear algebra and partial differential equations that are most useful to the life sciences are introduced as needed, and in the context of life science applications, are drawn from real, published papers. It also teaches students how to recognize when differential equations can help focus research. A course taught with this book can replace the standard course in multivariable calculus that is more usually suited to engineers and physicists.

The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44)

Author: John W. Morgan
Publisher: Princeton University Press
ISBN: 1400865166
Category: Mathematics
Page: 130
View: 9453

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The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Riemannian Geometry

Author: Sylvestre Gallot,Dominique Hulin,Jacques Lafontaine
Publisher: Springer Science & Business Media
ISBN: 364297242X
Category: Mathematics
Page: 286
View: 4024

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This book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.

The Ricci Flow in Riemannian Geometry

A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem
Author: Ben Andrews,Christopher Hopper
Publisher: Springer Science & Business Media
ISBN: 3642162851
Category: Mathematics
Page: 296
View: 9808

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Focusing on Hamilton's Ricci flow, this volume begins with a detailed discussion of the required aspects of differential geometry. The discussion also includes existence and regularity theory, compactness theorems for Riemannian manifolds, and much more.

Analysis, Manifolds, and Physics

Author: Yvonne Choquet-Bruhat,Cécile DeWitt-Morette,Margaret Dillard Bleick
Publisher: Gulf Professional Publishing
ISBN: 9780444860170
Category: Mathematics
Page: 630
View: 682

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This reference book, which has found wide use as a text, provides an answer to the needs of graduate physical mathematics students and their teachers. The present edition is a thorough revision of the first, including a new chapter entitled ``Connections on Principle Fibre Bundles'' which includes sections on holonomy, characteristic classes, invariant curvature integrals and problems on the geometry of gauge fields, monopoles, instantons, spin structure and spin connections. Many paragraphs have been rewritten, and examples and exercises added to ease the study of several chapters. The index includes over 130 entries.

Foundations of Differential Geometry, 2 Volume Set

Author: Shoshichi Kobayashi,Katsumi Nomizu
Publisher: Wiley
ISBN: 9780470555583
Category: Mathematics
Page: 832
View: 5534

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This set features: Foundations of Differential Geometry, Volume 1 (978-0-471-15733-5) and Foundations of Differential Geometry, Volume 2 (978-0-471-15732-8), both by Shoshichi Kobayashi and Katsumi Nomizu This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics. It is completely self-contained and will serve as a reference as well as a teaching guide. Volume 1 presents a systematic introduction to the field from a brief survey of differentiable manifolds, Lie groups and fibre bundles to the extension of local transformations and Riemannian connections. Volume 2 continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.

The Enlightenment: A Very Short Introduction

Author: John Robertson
Publisher: OUP Oxford
ISBN: 0191665134
Category: Philosophy
Page: 144
View: 9788

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A foundational moment in the history of modern European thought, the Enlightenment continues to be a reference point for philosophers, scholars and opinion-formers. To many it remains the inspiration of our commitments to the betterment of the human condition. To others, it represents the elevation of one set of European values to the world, many of whose peoples have quite different values. But what is the relationship between the historical Enlightenment and the idea of 'Enlightenment', and can these two understandings be reconciled? In this Very Short Introduction, John Robertson offers a concise historical introduction to the Enlightenment as an intellectual movement of eighteenth-century Europe. Discussing its intellectual achievements, he also explores how its supporters exploited new ways of communicating their ideas to a wider public, creating a new 'public sphere' for critical discussion of the moral, economic and political issues facing their societies. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

The Many Facets of Geometry

A Tribute to Nigel Hitchin
Author: Nigel J. Hitchin,Oscar Garcia-Prada,Jean Pierre Bourguignon,Simon Salamon
Publisher: Oxford University Press
ISBN: 0199534926
Category: Mathematics
Page: 434
View: 2789

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Few people have proved more influential in the field of differential and algebraic geometry, and in showing how this links with mathematical physics, than Nigel Hitchin. Oxford University's Savilian Professor of Geometry has made fundamental contributions in areas as diverse as: spin geometry, instanton and monopole equations, twistor theory, symplectic geometry of moduli spaces, integrables systems, Higgs bundles, Einstein metrics, hyperkähler geometry,Frobenius manifolds, Painlevé equations, special Lagrangian geometry and mirror symmetry, theory of grebes, and many more. The chapters in this fascinating volume, written by some of the greats in their fields (including four Fields Medalists), show how Hitchin's ideas have impacted on a wide variety ofsubjects. The book grew out of the Geometry Conference in Honour of Nigel Hitchin, held in Madrid, with some additional contributions, and should be required reading for anyone seeking insights into the overlap between geometry and physics.

Algebraic Models in Geometry

Author: Yves Félix,John Oprea,Daniel Tanré
Publisher: Oxford University Press
ISBN: 0199206511
Category: Mathematics
Page: 460
View: 8806

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In the past century, different branches of mathematics have become more widely separated. Yet, there is an essential unity to mathematics which still springs up in fascinating ways to solve interdisciplinary problems. This text provides a bridge between the subjects of algebraic topology, including differential topology, and geometry. It is a survey book dedicated to a large audience of researchers and graduate students in these areas. Containing a generalintroduction to the algebraic theory of rational homotopy and giving concrete applications of algebraic models to the study of geometrical problems, mathematicians in many areas will find subjects that are of interest to them in the book.

Geometry and Physics: Volume 2

A Festschrift in Honour of Nigel Hitchin
Author: Andrew Dancer,Jørgen Ellegaard Andersen,Oscar García-Prada
Publisher: N.A
ISBN: 0198802021
Page: 352
View: 5353

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These texts contain 29 articles that cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics

An Introduction to Differential Geometry

Author: T. J. Willmore
Publisher: Courier Corporation
ISBN: 0486282104
Category: Mathematics
Page: 336
View: 7393

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This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.

Comparison Geometry

Author: Karsten Grove,Peter Petersen
Publisher: Cambridge University Press
ISBN: 9780521592222
Category: Mathematics
Page: 262
View: 7514

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This is an up to date work on a branch of Riemannian geometry called Comparison Geometry.

An Introduction to Riemannian Geometry

With Applications to Mechanics and Relativity
Author: Leonor Godinho,José Natário
Publisher: Springer
ISBN: 3319086669
Category: Mathematics
Page: 467
View: 6973

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Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

Information Geometry

Near Randomness and Near Independence
Author: Khadiga Arwini,C. T. J. Dodson
Publisher: Springer
ISBN: 3540693939
Category: Mathematics
Page: 260
View: 6312

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This volume uses information geometry to give a common differential geometric framework for a wide range of illustrative applications including amino acid sequence spacings, cryptology studies, clustering of communications and galaxies, and cosmological voids.

Geometric and Algebraic Topological Methods in Quantum Mechanics

Author: G. Giachetta,L. Mangiarotti,Gennadii Aleksandrovich Sardanashvili
Publisher: World Scientific
ISBN: 9812701265
Category: Science
Page: 703
View: 7135

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In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry''s geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

The Ricci Flow

Techniques and Applications
Author: N.A
Publisher: American Mathematical Soc.
Category: Global differential geometry
Page: 517
View: 6305

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Entropy, $\mu$-invariant, and finite time singularities Geometric tools and point picking methods Geometric properties of $\kappa$-solutions Compactness of the space of $\kappa$-solutions Perelman's pseudolocality theorem Tools used in proof of pseudolocality Heat kernel for static metrics Heat kernel for evolving metrics Estimates of the heat equation for evolving metrics Bounds for the heat kernel for evolving metrics Elementary aspects of metric geometry Convex functions on Riemannian manifolds Asymptotic cones and Sharafutdinov retraction Solutions to selected exercises Bibliography Index