Publisher: PHI Learning Pvt. Ltd.

ISBN: 8120346505

Category: Mathematics

Page: 256

View: 4399

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, while trying to answer them using calculus techniques. The geometry of differentiable manifolds with structures is one of the most important branches of modern differential geometry. This well-written book discusses the theory of differential and Riemannian manifolds to help students understand the basic structures and consequent developments. While introducing concepts such as bundles, exterior algebra and calculus, Lie group and its algebra and calculus, Riemannian geometry, submanifolds and hypersurfaces, almost complex manifolds, etc., enough care has been taken to provide necessary details which enable the reader to grasp them easily. The material of this book has been successfully tried in classroom teaching. The book is designed for the postgraduate students of Mathematics. It will also be useful to the researchers working in the field of differential geometry and its applications to general theory of relativity and cosmology, and other applied areas. KEY FEATURES  Provides basic concepts in an easy-to-understand style.  Presents the subject in a natural way.  Follows a coordinate-free approach.  Includes a large number of solved examples and illuminating illustrations.  Gives notes and remarks at appropriate places.

Differential Geometry of Manifolds

Author: Stephen T. Lovett

Publisher: CRC Press

ISBN: 1439865469

Category: Mathematics

Page: 440

View: 5083

From the coauthor of Differential Geometry of Curves and Surfaces, this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices provide background information on point set topology, calculus of variations, and multilinear algebra—topics that may not have been covered in the prerequisite courses of multivariable calculus and linear algebra. Differential Geometry of Manifolds takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics, including the Hamiltonian formulation of dynamics (with a view toward symplectic manifolds), the tensorial formulation of electromagnetism, some string theory, and some fundamental concepts in general relativity.

Manifolds and Differential Geometry

Author: Jeffrey Marc Lee

Publisher: American Mathematical Soc.

ISBN: 0821848151

Category: Mathematics

Page: 671

View: 666

Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hyper-surfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.

Differential Geometry and Topology

Author: R Caddeo,F Tricerri

Publisher: World Scientific

ISBN: 9814553085


Page: 276

View: 7761

This volume contains the courses and lectures given during the workshop on Differential Geometry and Topology held at Alghero, Italy, in June 1992. The main goal of this meeting was to offer an introduction in attractive areas of current research and to discuss some recent important achievements in both the fields. This is reflected in the present book which contains some introductory texts together with more specialized contributions. The topics covered in this volume include circle and sphere packings, 3-manifolds invariants and combinatorial presentations of manifolds, soliton theory and its applications in differential geometry, G-manifolds of low cohomogeneity, exotic differentiable structures on R4, conformal deformation of Riemannian manifolds and Riemannian geometry of algebraic manifolds. Contents:Asystatic G-Manifolds (A Alekseevsky & D Alekseevsky)Les Paquets de Cercles (M Berger)Smooth Structures on Euclidean Spaces (S Demichelis)Surface Theory, Harmonic Maps and Commuting Hamiltonian Flows (D Ferus)Metric Invariants of Kähler Manifolds (M Gromov)On the Sphere Packing Problem and the Proof of Kepler's Conjecture (W Y Hsiang)A 3-Gem Approach to Turaev-Viro Invariants (S L S Lins)Cohomology Operations and Modular Invariant Theory (L Lomonaco)Scalar Curvature and Conformal Deformation of Riemannian Manifolds (A Ratto)Lectures on Combinatorial Presentations of Manifolds (O Viro) Readership: Mathematicians. keywords:

Lectures on the Geometry of Manifolds

Author: Liviu I. Nicolaescu

Publisher: World Scientific

ISBN: 9789810228361

Category: Mathematics

Page: 481

View: 4273

The object of this book is to introduce the reader to some of the most important techniques of modern global geometry. In writing it we had in mind the beginning graduate student willing to specialize in this very challenging field of mathematics. The necessary prerequisite is a good knowledge of the calculus with several variables, linear algebra and some elementary point-set topology.We tried to address several issues. 1. The Language; 2. The Problems; 3. The Methods; 4. The Answers.Historically, the problems came first, then came the methods and the language while the answers came last. The space constraints forced us to change this order and we had to painfully restrict our selection of topics to be covered. This process always involves a loss of intuition and we tried to balance this by offering as many examples and pictures as often as possible. We test most of our results and techniques on two basic classes examples: surfaces (which can be easily visualized) and Lie groups (which can be elegantly algebraized). When possible we present several facets of the same issue.We believe that a good familiarity with the formalism of differential geometry is absolutely necessary in understanding and solving concrete problems and this is why we presented it in some detail. Every new concept is supported by concrete examples interesting not only from an academic point of view.Our interest is mainly in global questions and in particular the interdependencegeometry ? topology, local ? global.We had to develop many algebraico-topological techniques in the special context of smooth manifolds. We spent a big portion of this book discussing the DeRham cohomology and its ramifications: Poincar‚ duality, intersection theory, degree theory, Thom isomorphism, characteristic classes, Gauss-Bonnet etc. We tried to calculate the cohomology groups of as many as possible concrete examples and we had to do this without relying on the powerful apparatus of homotopy theory (CW-complexes etc.). Some of the proofs are not the most direct ones but the means are sometimes more interesting than the ends. For example in computing the cohomology of complex grassmannians we returned to classical invariant theory and used some brilliant but unadvertised old ideas.In the last part of the book we discuss elliptic partial differential equations. This requires a familiarity with functional analysis. We painstakingly described the proofs of elliptic Lp and H”lder estimates (assuming some deep results of harmonic analysis) for arbitrary elliptic operators with smooth coefficients. It is not a ?light meal? but the ideas are useful in a large number of instances. We present a few applications of these techniques (Hodge theory, uniformization theorem). We conclude with a close look to a very important class of elliptic operators namely the Dirac operators. We discuss their algebraic structure in some detail, Weizenb”ck formul‘ and many concrete examples.

Differential Geometry Of Warped Product Manifolds And Submanifolds

Author: Chen Bang-yen

Publisher: World Scientific

ISBN: 9813208945

Category: Mathematics

Page: 516

View: 3039

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson–Walker models, are warped product manifolds. The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson–Walker's and Schwarzschild's. The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century. The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.

Fundamentals of Differential Geometry

Author: Serge Lang

Publisher: Springer Science & Business Media

ISBN: 1461205417

Category: Mathematics

Page: 540

View: 2262

This book provides an introduction to the basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas. This new edition includes new chapters, sections, examples, and exercises. From the reviews: "There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books." --EMS NEWSLETTER

Differential Geometry of Manifolds

Author: Uday Chand De,Absos Ali Shaikh

Publisher: Alpha Science International, Limited

ISBN: 9781842653715

Category: Mathematics

Page: 298

View: 8339

"Differential Geometry of Manifolds discusses the theory of differentiable and Riemannian manifolds to help students understand the basic structures and consequent developments. Since the tangent vector plays a crucial role in the study of differentiable manifolds, this idea has been thoroughly discussed. In the theory of Riemannian geometry some new proofs have been included to enable the reader understand the subject in a comprehensive and systematic manner." "This book will also benefit the postgraduate students as well as researchers working in the field of differential geometry and its applications to general relativity and cosmology."--BOOK JACKET.

Differential Geometry

Author: J. J. Stoker

Publisher: John Wiley & Sons

ISBN: 1118165470

Category: Mathematics

Page: 432

View: 9003

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.

Differential Geometry in Array Processing

Author: Athanassios Manikas

Publisher: Imperial College Press

ISBN: 9781860944239

Category: Mathematics

Page: 218

View: 5250

In view of the significance of the array manifold in array processing and array communications, the role of differential geometry as an analytical tool cannot be overemphasized. Differential geometry is mainly confined to the investigation of the geometric properties of manifolds in three-dimensional Euclidean space R3 and in real spaces of higher dimension.Extending the theoretical framework to complex spaces, this invaluable book presents a summary of those results of differential geometry which are of practical interest in the study of linear, planar and three-dimensional array geometries.

A Course in Differential Geometry

Author: Thierry Aubin

Publisher: American Mathematical Soc.

ISBN: 9780821872147

Category: Mathematics

Page: 184

View: 8184

This textbook for second-year graduate students is an introduction to differential geometry with principal emphasis on Riemannian geometry. The author is well-known for his significant contributions to the field of geometry and PDEs - particularly for his work on the Yamabe problem - and for his expository accounts on the subject. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.

Differential Geometry

Curves - Surfaces - Manifolds

Author: Wolfgang Kühnel

Publisher: American Mathematical Soc.

ISBN: 9780821839881

Category: Mathematics

Page: 380

View: 1548

Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in I\!\!R^3 that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.

Synthetic Geometry of Manifolds

Author: Anders Kock

Publisher: Cambridge University Press

ISBN: 0521116732

Category: Mathematics

Page: 302

View: 1708

This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighborhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.

Complex Differential Geometry

Author: Fangyang Zheng

Publisher: American Mathematical Soc.

ISBN: 0821829602

Category: Mathematics

Page: 264

View: 8195

The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds.The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classification theory, providing readers with some concrete examples of complex manifolds. The last part is the main purpose of the book; in it, the author discusses metrics, connections, curvature, and the various roles they play in the study of complex manifolds. A significant amount of exercises are provided to enhance student comprehension and practical experience.

Geometry of Differential Forms

Author: Shigeyuki Morita

Publisher: American Mathematical Soc.

ISBN: 9780821810453

Category: Mathematics

Page: 321

View: 5239

Since the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. This book is a comprehensive introduction to differential forms. It begins with a quick presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results about them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated in the book is a detailed description of the Chern-Weil theory. With minimal prerequisites, the book can serve as a textbook for an advanced undergraduate or a graduate course in differential geometry.

Foundations of Differential Geometry

Author: Shoshichi Kobayashi,Katsumi Nomizu

Publisher: University of Texas Press

ISBN: 9780471157335

Category: Mathematics

Page: 344

View: 3467

One of two volumes which lay the foundations for understanding differential geometry. This work familiarizes readers with various techniques of computation.

Differential Geometry of Curves and Surfaces

Author: Masaaki Umehara,Kotaro Yamada

Publisher: World Scientific Publishing Company

ISBN: 9814740268


Page: 328

View: 4961

This engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well. Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates. Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities. In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field. Request Inspection Copy

Partial Differential Equations on Manifolds

Author: Robert Everist Greene,Shing-Tung Yau

Publisher: American Mathematical Soc.

ISBN: 082181494X

Category: Mathematics

Page: 560

View: 6443

The first of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Part 1 begins with a problem list by S.T. Yau, successor to his 1980 list ( Sem

Differentialgeometrie von Kurven und Flächen

Author: Manfredo P. do Carmo

Publisher: Springer-Verlag

ISBN: 3322850722

Category: Technology & Engineering

Page: 263

View: 2644

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

Geometry of Vector Sheaves

An Axiomatic Approach to Differential Geometry

Author: Anastasios Mallios

Publisher: Springer Science & Business Media

ISBN: 9780792350057

Category: Geometry, Differential

Page: 436

View: 687

This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasized. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology ("differential spaces"), to non-linear PDEs (generalised functions). Intended for postgraduate students and researchers whose work involves differential geometry, global analysis, analysis on manifolds, algebraic topology, sheaf theory, cohomology, functional analysis or abstract harmonic analysis.