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Author : John McCleary
Publisher : Cambridge University Press
Page : 375 pages
File Size : 51,5 Mb
Release : 2013
Category : Mathematics
ISBN : 9780521116077
A thoroughly revised second edition of a textbook for a first course in differential/modern geometry that introduces methods within a historical context.
Author : John Willard Milnor,David W. Weaver
Publisher : Princeton University Press
Page : 80 pages
File Size : 53,5 Mb
Release : 1997-12-14
Category : Mathematics
ISBN : 0691048339
This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.
Author : Morris W. Hirsch
Publisher : Springer Science & Business Media
Page : 230 pages
File Size : 46,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468494495
"A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology....There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text." —MATHEMATICAL REVIEWS
Author : Torsten Wedhorn
Publisher : Springer
Page : 366 pages
File Size : 49,7 Mb
Release : 2016-07-25
Category : Mathematics
ISBN : 9783658106331
This book explains techniques that are essential in almost all branches of modern geometry such as algebraic geometry, complex geometry, or non-archimedian geometry. It uses the most accessible case, real and complex manifolds, as a model. The author especially emphasizes the difference between local and global questions. Cohomology theory of sheaves is introduced and its usage is illustrated by many examples.
Author : Victor Guillemin,Alan Pollack
Publisher : American Mathematical Soc.
Page : 242 pages
File Size : 47,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821851937
Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.
Author : Christian Bär
Publisher : Cambridge University Press
Page : 335 pages
File Size : 48,9 Mb
Release : 2010-05-06
Category : Mathematics
ISBN : 9780521896719
This easy-to-read introduction takes the reader from elementary problems through to current research. Ideal for courses and self-study.
Author : Joel W. Robbin,Dietmar A. Salamon
Publisher : Springer Nature
Page : 426 pages
File Size : 45,7 Mb
Release : 2022-01-12
Category : Mathematics
ISBN : 9783662643402
This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.
Author : Norbert A'Campo
Publisher : Springer Nature
Page : 282 pages
File Size : 47,7 Mb
Release : 2021-10-27
Category : Mathematics
ISBN : 9783030890322
This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.
Author : John M. Lee
Publisher : Springer Science & Business Media
Page : 646 pages
File Size : 45,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9780387217529
Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why
Author : Raoul Bott,Loring W. Tu
Publisher : Springer Science & Business Media
Page : 319 pages
File Size : 47,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475739510
Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.
Author : C. G. Gibson
Publisher : Cambridge University Press
Page : 236 pages
File Size : 42,5 Mb
Release : 2001-05-17
Category : Mathematics
ISBN : 0521011078
This book is an introductory text on the differential geometry of plane curves.
Author : Shigeyuki Morita
Publisher : American Mathematical Soc.
Page : 356 pages
File Size : 55,7 Mb
Release : 2001
Category : Mathematics
ISBN : 0821810456
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. The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning 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 is a detailed description of the Chern-Weil theory. The book can serve as a textbook for undergraduate students and for graduate students in geometry.
Author : S. Ramanan
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 49,8 Mb
Release : 2005
Category : Analytic spaces
ISBN : 9780821837023
The power that analysis, topology and algebra bring to geometry has revolutionised the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on Global Analysis.
Author : John M. Lee
Publisher : Springer Science & Business Media
Page : 395 pages
File Size : 43,7 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387227276
Manifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces.
Author : Barrett O'Neill
Publisher : Academic Press
Page : 422 pages
File Size : 42,9 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483268118
Elementary Differential Geometry focuses on the elementary account of the geometry of curves and surfaces. The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation. The text is a valuable reference for students interested in elementary differential geometry.
