Differential Geometry

Bundles, Connections, Metrics and Curvature
Author: Clifford Henry Taubes
Publisher: OUP Oxford
ISBN: 0191621226
Category: Mathematics
Page: 312
View: 4685

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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: 8182

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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: 2962

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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.

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: 6345

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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.

Geometry and Physics: Volume II

A Festschrift in honour of Nigel Hitchin
Author: Andrew Dancer,Jørgen Ellegaard Andersen,Oscar García-Prada
Publisher: Oxford University Press
ISBN: 019252237X
Category: Mathematics
Page: 400
View: 1291

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Nigel Hitchin is one of the world's foremost figures in the fields of differential and algebraic geometry and their relations with mathematical physics, and he has been Savilian Professor of Geometry at Oxford since 1997. Geometry and Physics: A Festschrift in honour of Nigel Hitchin contain the proceedings of the conferences held in September 2016 in Aarhus, Oxford, and Madrid to mark Nigel Hitchin's 70th birthday, and to honour his far-reaching contributions to geometry and mathematical physics. These texts contain 29 articles by contributors to the conference and other distinguished mathematicians working in related areas, including three Fields Medallists. The articles 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.

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
Category:
Page: 352
View: 9475

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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

Hilbert


Author: Constance Reid,Hermann Weyl
Publisher: Springer-Verlag
ISBN: 3662286157
Category: Mathematics
Page: 290
View: 7444

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Vektoranalysis


Author: Klaus Jänich
Publisher: Springer-Verlag
ISBN: 3662107503
Category: Mathematics
Page: 277
View: 6360

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Die Vektoranalysis handelt, in klassischer Darstellung, von Vektorfeldern, den Operatoren Gradient, Divergenz und Rotation, von Linien-, Flächen- und Volumenintegralen und von den Integralsätzen von Gauß, Stokes und Green. In moderner Fassung ist es der Cartansche Kalkül mit dem Satz von Stokes. Das vorliegende Buch vertritt grundsätzlich die moderne Herangehensweise, geht aber auch sorgfältig auf die klassische Notation und Auffassung ein. Das Buch richtet sich an Mathematik- und Physikstudenten ab dem zweiten Studienjahr, die mit den Grundbegriffen der Differential- und Integralrechnung in einer und mehreren Variablen sowie der Topologie vertraut sind. Der sehr persönliche Stil des Autors und die aus anderen Büchern bereits bekannten Lernhilfen, wie: viele Figuren, mehr als 50 kommentierte Übungsaufgaben, über 100 Tests mit Antworten, machen auch diesen Text zum Selbststudium hervorragend geeignet.

Differentialgeometrie, Topologie und Physik


Author: Mikio Nakahara
Publisher: Springer-Verlag
ISBN: 3662453002
Category: Science
Page: 597
View: 7566

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Differentialgeometrie und Topologie sind wichtige Werkzeuge für die Theoretische Physik. Insbesondere finden sie Anwendung in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das vorliegende beliebte Buch, das nun erstmals ins Deutsche übersetzt wurde, ist eine ideale Einführung für Masterstudenten und Forscher im Bereich der theoretischen und mathematischen Physik. - Im ersten Kapitel bietet das Buch einen Überblick über die Pfadintegralmethode und Eichtheorien. - Kapitel 2 beschäftigt sich mit den mathematischen Grundlagen von Abbildungen, Vektorräumen und der Topologie. - Die folgenden Kapitel beschäftigen sich mit fortgeschritteneren Konzepten der Geometrie und Topologie und diskutieren auch deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie. - Daran anschließend findet eine Zusammenführung von Geometrie und Topologie statt: es geht um Faserbündel, characteristische Klassen und Indextheoreme (u.a. in Anwendung auf die supersymmetrische Quantenmechanik). - Die letzten beiden Kapitel widmen sich der spannendsten Anwendung von Geometrie und Topologie in der modernen Physik, nämlich den Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer gemetrischen Perspektive. Mikio Nakahara studierte an der Universität Kyoto und am King’s in London Physik sowie klassische und Quantengravitationstheorie. Heute ist er Physikprofessor an der Kinki-Universität in Osaka (Japan), wo er u. a. über topologische Quantencomputer forscht. Diese Buch entstand aus einer Vorlesung, die er während Forschungsaufenthalten an der University of Sussex und an der Helsinki University of Sussex gehalten hat.

Partielle Differentialgleichungen

Eine Einführung
Author: Walter A. Strauss
Publisher: Springer-Verlag
ISBN: 366312486X
Category: Mathematics
Page: 458
View: 9015

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Dieses Buch ist eine umfassende Einführung in die klassischen Lösungsmethoden partieller Differentialgleichungen. Es wendet sich an Leser mit Kenntnissen aus einem viersemestrigen Grundstudium der Mathematik (und Physik) und legt seinen Schwerpunkt auf die explizite Darstellung der Lösungen. Es ist deshalb besonders auch für Anwender (Physiker, Ingenieure) sowie für Nichtspezialisten, die die Methoden der mathematischen Physik kennenlernen wollen, interessant. Durch die große Anzahl von Beispielen und Übungsaufgaben eignet es sich gut zum Gebrauch neben Vorlesungen sowie zum Selbststudium.

Differentialgeometrie von Kurven und Flächen


Author: Manfredo P. do Carmo
Publisher: Springer-Verlag
ISBN: 3322850722
Category: Technology & Engineering
Page: 263
View: 3152

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Inhalt: Kurven - Reguläre Flächen - Die Geometrie der Gauß-Abbildung - Die innere Geometrie von Flächen - Anhang

Special Relativity


Author: Nicholas M.J. Woodhouse
Publisher: Springer-Verlag
ISBN: 3540466762
Category: Science
Page: 88
View: 7028

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Differentialgeometrie

Kurven - Flächen - Mannigfaltigkeiten
Author: Wolfgang Kühnel
Publisher: Springer-Verlag
ISBN: 3658006153
Category: Mathematics
Page: 284
View: 9806

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Dieses Buch ist eine Einführung in die Differentialgeometrie und ein passender Begleiter zum Differentialgeometrie-Modul (ein- und zweisemestrig). Zunächst geht es um die klassischen Aspekte wie die Geometrie von Kurven und Flächen, bevor dann höherdimensionale Flächen sowie abstrakte Mannigfaltigkeiten betrachtet werden. Die Nahtstelle ist dabei das zentrale Kapitel "Die innere Geometrie von Flächen". Dieses führt den Leser bis hin zu dem berühmten Satz von Gauß-Bonnet, der ein entscheidendes Bindeglied zwischen lokaler und globaler Geometrie darstellt. Die zweite Hälfte des Buches ist der Riemannschen Geometrie gewidmet. Den Abschluss bildet ein Kapitel über "Einstein-Räume", die eine große Bedeutung sowohl in der "Reinen Mathematik" als auch in der Allgemeinen Relativitätstheorie von A. Einstein haben. Es wird großer Wert auf Anschaulichkeit gelegt, was durch zahlreiche Abbildungen unterstützt wird. Bei der Neuauflage wurden einige zusätzliche Lösungen zu den Übungsaufgaben ergänzt.

Pascal, Fermat und die Berechnung des Glücks

eine Reise in die Geschichte der Mathematik
Author: Keith J. Devlin
Publisher: C.H.Beck
ISBN: 9783406590993
Category: Wahrscheinlichkeitstheorie - Geschichte
Page: 204
View: 8672

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Der Autor stellt die Entstehung der Wahrscheinlichkeitsrechnung und den damit verbundenen Wandel des menschlichen Alltagslebens dar.

Grundzüge der Mengenlehre


Author: Felix Hausdorff
Publisher: American Mathematical Soc.
ISBN: 9780828400619
Category: Mathematics
Page: 476
View: 7536

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This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.

Moderne Physik


Author: Paul Allen Tipler,Ralph A. Llewellyn
Publisher: Oldenbourg Verlag
ISBN: 9783486255645
Category: Physics
Page: 946
View: 2571

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Die Autoren zeigen, dass man ein tiefgehendes Verstandnis der Modernen Physik vermitteln kann, ohne einen schwerfalligen mathematischen Apparat bemuhen zu mussen. Mit uber 500 Abbildungen, Zitaten beruhmter Physiker sowie mit ca. 700 sorgfaltig ausgewahlten Ubungsaufgaben und uber das Internet zuganglichen Erganzungen wurde die "Modern Physics" in den USA zu einem der beliebtesten Lehrbucher zu diesem Thema."

Allgemeine Flächentheorie

(Disquisitiones Generales Circa Superficies Curvas)
Author: Carl Friedrich Gauss,A. Wangerin
Publisher: N.A
ISBN: 9780649765799
Category:
Page: 74
View: 9450

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