De Rham cohomology is the cohomology of differential forms. This book offers a self-contained exposition to this subject and to the theory of characteristic classes from the curvature point of view. It requires no prior knowledge of the concepts of algebraic topology or cohomology. The first ten chapters study cohomology of open sets in Euclidean space, treat smooth manifolds and their cohomology and end with integration on manifolds. The last eleven chapters cover Morse theory, index of vector fields, Poincaré duality, vector bundles, connections and curvature, Chern and Euler classes, Thom isomorphism, and the general Gauss-Bonnet theorem. The text includes over 150 exercises, and gives the background necessary for the modern developments in gauge theory and geometry in four dimensions, but it also serves as an introductory course in algebraic topology. It will be invaluable to anyone who wishes to know about cohomology, curvature, and their applications.

In Hypo-Analytic Structures Franois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometric entities attached to those structures are isolated, such as maximally real submanifolds and orbits of the system. Treves discusses the existence, uniqueness, and approximation of local solutions to homogeneous and inhomogeneous equations and delimits their supports. The contents of this book consist of many results accumulated in the last decade by the author and his collaborators, but also include classical results, such as the Newlander-Nirenberg theorem. The reader will find an elementary description of the FBI transform, as well as examples of its use. Treves extends the main approximation and uniqueness results to first-order nonlinear equations by means of the Hamiltonian lift.

Originally published in 1993.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In der Reihe "TEUBNER-ARCHIV zur Mathematik" werden bedeutende klassische Arbeiten kommentiert, mit aktuellen Anmerkungen versehen und durch Literaturhin- weise ergiinzt. Dieser erste Band enthillt fotomechanische Nachdrucke von vier Beitragen der Mathe- matiker C. F. GAUSS, B. RIEMANN und H. MINKOWSKI. Diese Arbeiten waren grund- legend filr die Entwicklung und Weiterentwicklung der Differentialgeometrie als innere Geometrie bis zur allgemeinen Rel, ativitatstheorie. Es ist gewiB nicht nur ein Zufall, daB sich filr diese drei Manner die produktive Zeit des Wirkens auf dem genannten Gebiet der Geometrie in der Universitiitsstadt Gottingen vollzog. Durch die folgenden Satze ALBERT EINSTEINS aus seiner Abhandlung tiber die Grund- ztige der Relativitatstheorie aus dem Jahre 1922 lassen sich in einfacher und klarer Weise die diesbeztiglichen Verdienste dieser drei Mathematiker charakterisieren: "GAUSS hat in seiner Fliichentheorie die metrischen Eigenschaften einer in einem dreidimensionalen euklidischen Raum eingebetteten Fliiche untersucht und gezeigt, daB diese durch Begriffe beschrieben werden konnen, die sich nur auf die Flache selbst, nicht aber auf die Ein- bettung beziehen . . . RIEMANN dehnte den GauBschen Gedankengang auf Kontinua beliebiger Dimensionszahl aus; er hat die physikalische Bedeutung dieser Verallgemei- nerung der Geometrie EUKLIDS mit prophetischem Blick vorausgesehen . . . Durch die Einfilhrung der imaginiiren Zeitvariable X4 = it hat MINKOWSKI die Invariantentheorie des vierdimensionalen Kontinuums des physikalischen Geschehens der des dreidimen- sionalen Kontinuums des euklidischen Raumes vollig analog gemacht.

Aus dem Vorwort: "Globale Probleme der Differentialgeometrie erfreuen sich eines immer noch wachsenden Interesses. Gerade in der Riemannschen Geometrie hat die Frage nach Beziehungen zwischen Riemannscher und topologischer Struktur in neuerer Zeit zu vielen schonen und uberraschenden Einsichten gefuhrt. Dabei denken wir hier vor allem an den Problemkreis: Welche topologischen Invarianten werden charakterisiert durch eine der wichtigsten isometrischen Invarianten, die Krummung? Ziel der folgenden Noten ist, einige zentrale Resultate in dieser Richtung darzustellen.... Wir haben uns bemuht, die Darstellung moglichst elementar und in sich abgeschlossen zu halten und einen einfachen leistungsfahigen Kalkul zu entwickeln.""

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrodinger operators on the left hand side and a critical nonlinearity on the right hand side.

A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.

Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields."

This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.

This book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic.

Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfangen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen fur die historische wie auch die disziplingeschichtliche Forschung zur Verfugung, die jeweils im historischen Kontext betrachtet werden mussen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Differential Geometry in Physics is a treatment of the mathematical foundations of the theory of general relativity and gauge theory of quantum fields. The material is intended to help bridge the gap that often exists between theoretical physics and applied mathematics. The approach is to carve an optimal path to learning this challenging field by appealing to the much more accessible theory of curves and surfaces. The transition from classical differential geometry as developed by Gauss, Riemann and other giants, to the modern approach, is facilitated by a very intuitive approach that sacrifices some mathematical rigor for the sake of understanding the physics. The book features numerous examples of beautiful curves and surfaces often reflected in nature, plus more advanced computations of trajectory of particles in black holes. Also embedded in the later chapters is a detailed description of the famous Dirac monopole and instantons. Features of this book: * Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years. * The material in the other chapters has served as the foundation for many master's thesis at University of North Carolina Wilmington for students seeking doctoral degrees. * An open access ebook edition is available at Open UNC (https: //openunc.org) * The book contains over 80 illustrations, including a large array of surfaces related to the theory of soliton waves that does not commonly appear in standard mathematical texts on differential geometry.

A sequel to Lectures on Riemann Surfaces (Mathematical Notes, 1966), this volume continues the discussion of the dimensions of spaces of holomorphic cross-sections of complex line bundles over compact Riemann surfaces. Whereas the earlier treatment was limited to results obtainable chiefly by one-dimensional methods, the more detailed analysis presented here requires the use of various properties of Jacobi varieties and of symmetric products of Riemann surfaces, and so serves as a further introduction to these topics as well. The first chapter consists of a rather explicit description of a canonical basis for the Abelian differentials on a marked Riemann surface, and of the description of the canonical meromorphic differentials and the prime function of a marked Riemann surface. Chapter 2 treats Jacobi varieties of compact Riemann surfaces and various subvarieties that arise in determining the dimensions of spaces of holomorphic cross-sections of complex line bundles. In Chapter 3, the author discusses the relations between Jacobi varieties and symmetric products of Riemann surfaces relevant to the determination of dimensions of spaces of holomorphic cross-sections of complex line bundles. The final chapter derives Torelli's theorem following A. Weil, but in an analytical context. Originally published in 1973. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Bei der Herausgabe dieses Buches mochte ich an dieser Stelle Herrn L. Berwald in Prag, Herrn D. J. Struik in Delft und Herrn R. WeitzenbOck in Blaricum, die mich durch das Mitlesen der Korrek turen sowie durch viele wichtige Bemerkungen aufs wirksamste unter stiitzt haben, meinen verbindlichsten Dank aussprechen. Einen freundschaftlichen GruB dem mathematischen Kreise in Hamburg, wo es mir vergonnt war, im Sommersemester dieses Jahres iiber die mehrdimensionale Affingeometrie zu lesen. Manche anregende Bemerkung zum vierten Abschnitt brachte mir diese schOne Zeit, die mir immer in freudiger Erinnerung bleiben wird. Der Verlagsbuchhandlung Julius Springer meinen besonderen Dank fiir die sorgfaltige Behandlung der Korrekturen, die mir die sauere Arbeit des Korrigierens fast zu einer Freude machte. Delft, im Dezember 1923. J. A. Schouten. Inhaltsverzeichnis. Seite Einleitung . . . 1 I. Der algebraische Tei des Kalkiils. 1. Die allgemeine Mannigfaltigkeit Xn . . 8 2. Der Begriff der Ubertragung . . . . . . 9 3. Die euklidischaffine Mannigfaltigkeit En . 9 4. Kontravariante und kovariante Vektoren . 12 5. Kontravariante und kovariante Bivektoren, Trivektoren usw. 17 6. Geometrische Darstellung kontravarianter und kovarianter p-Vektoren bei Einschrankung der Gruppe 20 7. Allgemeine GriiBen . . . . . . . . . . 23 8. Die Uberschiebungen . . . . . . . . . 28 9. Geometrische Darstellung der Tensoren 32 10. GriiBen zweiten Grades und lineare Transformationen 33 11. Die Einfiihrung einer MaBbestimmung in der En . . 36 12. Die Fundamentaltensoren. . . . . . . . . . . . . 38 13. Geometrische Darstellung alternierender GriiBen bei der orthogonalen und rotationalen Gruppe. Metrische Eigenschaften . . . . . . . . 41 14. Metrische Eigenschaften eines Te-nsors zweiten Grades. . . . . . ."

Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data. In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees. This volume is based on lectures delivered at the 2018 AMS Short Course ``Discrete Differential Geometry,'' held January 8-9, 2018, in San Diego, California. The papers in this volume illustrate the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.

Study 79 contains a collection of papers presented at the Conference on Discontinuous Groups and Ricmann Surfaces at the University of Maryland, May 21-25, 1973. The papers, by leading authorities, deal mainly with Fuchsian and Kleinian groups, Teichmuller spaces, Jacobian varieties, and quasiconformal mappings. These topics are intertwined, representing a common meeting of algebra, geometry, and analysis.