This publication is an creation to differential manifolds. It provides stable preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie thought. It presupposes little historical past: the reader is barely anticipated to grasp simple differential calculus, and a bit point-set topology. The booklet covers the most themes of differential geometry: manifolds, tangent area, vector fields, differential varieties, Lie teams, and some extra refined subject matters corresponding to de Rham cohomology, measure idea and the Gauss-Bonnet theorem for surfaces.
Its ambition is to offer good foundations. specifically, the creation of “abstract” notions akin to manifolds or differential varieties is influenced through questions and examples from arithmetic or theoretical physics. greater than a hundred and fifty routines, a few of them effortless and classical, a few others extra refined, may help the newbie in addition to the extra specialist reader. options are supplied for many of them.
The booklet may be of curiosity to varied readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to gather a few feeling approximately this pretty theory.
The unique French textual content creation aux variétés différentielles has been a best-seller in its class in France for lots of years.
Jacques Lafontaine was once successively assistant Professor at Paris Diderot college and Professor on the collage of Montpellier, the place he's almost immediately emeritus. His major learn pursuits are Riemannian and pseudo-Riemannian geometry, together with a few points of mathematical relativity. in addition to his own study articles, he was once inquisitive about a number of textbooks and learn monographs.
Das Buch bietet eine Einführung in die Topologie, Differentialtopologie und Differentialgeometrie. Es basiert auf Manuskripten, die in verschiedenen Vorlesungszyklen erprobt wurden. Im ersten Kapitel werden grundlegende Begriffe und Resultate aus der mengentheoretischen Topologie bereitgestellt. Eine Ausnahme hiervon bildet der Jordansche Kurvensatz, der für Polygonzüge bewiesen wird und eine erste Idee davon vermitteln soll, welcher artwork tiefere topologische Probleme sind. Im zweiten Kapitel werden Mannigfaltigkeiten und Liesche Gruppen eingeführt und an einer Reihe von Beispielen veranschaulicht. Diskutiert werden auch Tangential- und Vektorraumbündel, Differentiale, Vektorfelder und Liesche Klammern von Vektorfeldern. Weiter vertieft wird diese Diskussion im dritten Kapitel, in dem die de Rhamsche Kohomologie und das orientierte vital eingeführt und der Brouwersche Fixpunktsatz, der Jordan-Brouwersche Zerlegungssatz und die Integralformel von Stokes bewiesen werden. Das abschließende vierte Kapitel ist den Grundlagen der Differentialgeometrie gewidmet. Entlang der Entwicklungslinien, die die Geometrie der Kurven und Untermannigfaltigkeiten in Euklidischen Räumen durchlaufen hat, werden Zusammenhänge und Krümmung, die zentralen Konzepte der Differentialgeometrie, diskutiert. Den Höhepunkt bilden die Gaussgleichungen, die model des theorema egregium von Gauss für Untermannigfaltigkeiten beliebiger measurement und Kodimension.
Das Buch richtet sich in erster Linie an Mathematik- und Physikstudenten im zweiten und dritten Studienjahr und ist als Vorlage für ein- oder zweisemestrige Vorlesungen geeignet.
By Stefano Pigola
The purpose of this paper is to introduce the reader to varied kinds of the utmost precept, ranging from its classical formula as much as generalizations of the Omori-Yau greatest precept at infinity lately received by way of the authors. purposes are given to a couple of geometrical difficulties within the environment of whole Riemannian manifolds, lower than assumptions both at the curvature or at the quantity development of geodesic balls.
By Richard Schoen
This quantity provides lectures given through Richard Schoen and Shing-Tung Yau on the Institute for complicated experiences at Princeton college in 1984 and 1985. The lectures describe the foremost advances in differential geometry, which improved quickly within the 20th century.
By S. K. Donaldson
This monograph offers an intensive exposition of Floer's seminal paintings in the course of the Eighties from a modern point of view. the fabric contained right here was once built with particular purposes in brain. besides the fact that, it has now develop into transparent that the options used are very important for lots of present parts of analysis. an enormous instance will be symplectic concept and gluing difficulties for self-dual metrics and different metrics with distinct holonomy. the writer writes with the massive photograph continuously in brain. in addition to a assessment of the present kingdom of information, there are sections at the most likely course of destiny examine. integrated during this are connections among Floer teams and the prestigious Seiberg-Witten invariants. the consequences defined during this quantity shape a part of the realm often called Donaldson thought. the importance of this paintings is such that the writer was once offered the distinguished Fields Medal for his contribution.
By P.R. Halmos
From the Preface: "This ebook used to be written for the lively reader. the 1st half includes difficulties, often preceded by way of definitions and motivation, and infrequently by way of corollaries and old remarks... the second one half, a truly brief one, comprises hints... The 3rd half, the longest, comprises strategies: proofs, solutions, or contructions, reckoning on the character of the problem....
This isn't really an advent to Hilbert area conception. a few wisdom of that topic is a prerequisite: no less than, a examine of the weather of Hilbert area concept should still continue at the same time with the studying of this book."
Fibre bundles, now an essential component of differential geometry, also are of significant value in sleek physics--such as in gauge idea. This e-book, a succinct creation to the topic by means of renown mathematician Norman Steenrod, was once the 1st to provide the topic systematically.
It starts with a common creation to bundles, together with such issues as differentiable manifolds and masking areas. the writer then presents short surveys of complicated issues, comparable to homotopy idea and cohomology concept, sooner than utilizing them to review extra homes of fibre bundles. the result's a vintage and undying paintings of serious software that would attract critical mathematicians and theoretical physicists alike.
Riemannian Geometry is an improved variation of a hugely acclaimed and profitable textbook (originally released in Portuguese) for first-year graduate scholars in arithmetic and physics. The author's therapy is going very on to the elemental language of Riemannian geometry and instantly provides a few of its so much basic theorems. it really is hassle-free, assuming just a modest historical past from readers, making it appropriate for a wide selection of scholars and path buildings. Its number of subject matters has been deemed "superb" through academics who've used the textual content.
A major function of the e-book is its strong and revealing constitution, starting easily with the definition of a differentiable manifold and finishing with probably the most very important leads to Riemannian geometry, an evidence of the field Theorem. The textual content abounds with uncomplicated definitions and theorems, examples, purposes, and various workouts to check the student's knowing and expand wisdom and perception into the topic. teachers and scholars alike will locate the paintings to be a major contribution to this hugely acceptable and stimulating subject.
By David Bachman
This textual content offers differential varieties from a geometrical standpoint available on the undergraduate point. It starts with simple suggestions corresponding to partial differentiation and a number of integration and lightly develops the total equipment of differential varieties. the topic is approached with the concept complicated recommendations will be outfitted up through analogy from less complicated circumstances, which, being inherently geometric, frequently might be most sensible understood visually. every one new suggestion is gifted with a traditional photograph that scholars can simply seize. Algebraic houses then stick to. The publication includes first-class motivation, a number of illustrations and strategies to chose problems.
This is the 1st exposition of the quantization thought of singular symplectic (Marsden-Weinstein) quotients and their purposes to physics. The reader will collect an advent to many of the concepts utilized in this region, in addition to an outline of the newest study techniques. those contain classical differential and algebraic geometry, in addition to operator algebras and noncommutative geometry. hence one may be amply ready to stick to destiny advancements during this field.
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