Differential and Riemannian geometry Books
Dover Publications Inc. Differential Geometry
Book SynopsisAn introductory textbook on the differential geometry of curves and surfaces in 3-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures, and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods, and results involved. With problems and solutions. Includes 99 illustrations.
£14.39
Dover Publications Inc. Differential Geometry of Curves and Surfaces
Book SynopsisOne of the most widely used texts in its field, this volume has been continuously in print since its initial 1976 publication. The clear, well-written exposition is enhanced by many examples and exercises, some with hints and answers. Prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables.
£26.79
Princeton University Press Morse Theory
Book SynopsisOne classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. This book offers an exposition of Morse theory by John Milnor, recipient of the Fields Medal in 1962.Trade Review"John Milnor, Winner of the 2011 Abel Prize from the Norwegian Academy of Science and Letters""John Willard Milnor, Winner of the 2011 Leroy P. Steele Prize for Lifetime Achievement, American Mathematical Society"Table of Contents*Frontmatter, pg. i*PREFACE, pg. v*CONTENTS, pg. vii*PART I. NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD, pg. 1*PART II. A RAPID COURSE IN RIEMANNIAN GEOMETRY, pg. 43*PART III. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS, pg. 67*PART IV. APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES, pg. 109*APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION, pg. 149
£59.50
World Scientific Publishing Co Pte Ltd Introduction To Modern Finsler Geometry
Book SynopsisThis comprehensive book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds.In Part I, the authors discuss differential manifolds, Finsler metrics, the Chern connection, Riemannian and non-Riemannian quantities. Part II is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations, comparison theorems, fundamental group, minimal immersions, harmonic maps, Einstein metrics, conformal transformations, amongst other related topics. The authors made great efforts to ensure that the contents are accessible to senior undergraduate students, graduate students, mathematicians and scientists.
£38.00
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Collected Works
Book SynopsisWhile Eugenio Calabi is best known for his contributions to the theory of Calabi-Yau manifolds, this Steele-Prize-winning geometer’s fundamental contributions to mathematics have been far broader and more diverse than might be guessed from this one aspect of his work. His works have deep influence and lasting impact in global differential geometry, mathematical physics and beyond. By bringing together 47 of Calabi’s important articles in a single volume, this book provides a comprehensive overview of his mathematical oeuvre, and includes papers on complex manifolds, algebraic geometry, Kähler metrics, affine geometry, partial differential equations, several complex variables, group actions and topology. The volume also includes essays on Calabi’s mathematics by several of his mathematical admirers, including S.K. Donaldson, B. Lawson and S.-T. Yau, Marcel Berger; and Jean Pierre Bourguignon. This book is intended for mathematicians and graduate students around the world. Calabi’s visionary contributions will certainly continue to shape the course of this subject far into the future.Trade Review“In my case, I spent several happy hours learning about affine differential geometry, something that would certainly never have happened if I had not picked up this volume. … The collected works of Eugenio Calabi are worthy of a place on the bookshelf of any person with a serious interest in differential geometry.” (Joel Fine, EMS Magazine, May 11, 2023)Table of ContentsPreface.- J.-P. Bourguignon, Eugenio Calabi’s Short Biography.- Bibliographic List of Works.- S.-T. Yau, An Essay on Eugenio Calabi.- Part I: Commentaries on Calabi’s Life and Work: B. Lawson, Reflections on the Early Work of Eugenio Calabi.- M. Berger, Encounter with a Geometer: Eugenio Calabi.- J.-P. Bourguignon, Eugenio Calabi and Kähler Metrics.- C. LeBrun, Eugenio Calabi and the Curvature of Kähler Manifolds.- X. Chen, S. Donaldson, Calabi’s Work on Affine Differential Geometry and Results of Bernstein Type.- Part II: Collected Works: E. Calabi ,Ar. Dvoretzky, Convergence- and Sum-Factors for Series of Complex Numbers (1951).- E. Calabi, D. C. Spencer, Completely Integrable Almost Complex Manifolds (1951).- E. Calabi, Metric Riemann Surfaces (1953).- E. Calabi, M. Rosenlicht, Complex Analytic Manifolds Without Countable Base (1953).- E. Calabi, B. Eckmann, A Class of Compact, Complex Manifolds Which Are Not Algebraic (1953).- E. Calabi, Isometric Imbedding of Complex Manifolds (1953).- E. Calabi, The Space of Kähler Metrics (1954).- E. Calabi, The Variation of Kähler Metrics I. The Structure of the Space (1954).- E. Calabi, The Variation of Kähler Metrics II. A Minimum Problem (1954).- E. Calabi, On Kähler Manifolds With Vanishing Canonical Class (1957).- E. Calabi, Construction and Properties of Some 6-Dimensional Almost Complex Manifolds (1958).- E. Calabi, Improper Affine Hyperspheres of Convex Type and a Generalization of a Theorem by K. Jörgens (1958).- E. Calabi, An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry (1958).- E. Calabi, Errata: An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry (1959).- E. Calabi, E. Vesentini, Sur les variétés complexes compactes localement symétriques (1959).- E. Calabi, E. Vesentini, On Compact, Locally Symmetric Kähler Manifolds (1960).- E. Calabi, On Compact, Riemannian Manifolds with Constant Curvature I. (1961).- E. Calabi, L. Markus Relativistic Space Forms (1962).- E. Calabi, Linear Systems of Real Quadratic Forms (1964).- E. Calabi, Quasi-Surjective Mappings and a Generalization of Morse Theory (1966).- E. Calabi, Minimal Immersions of Surfaces in Euclidean Spheres (1967).- E. Calabi, On Ricci Curvature and Geodesics (1967).- E. Calabi, On Differentiable Actions of Compact Lie Groups on Compact Manifolds (1968).- E. Calabi, An Intrinsic Characterization of Harmonic One-Forms (1969).- E. Calabi, On the Group of Automorphisms of a Symplectic Manifold (1970).- E. Calabi, P. Hartman, On the Smoothness of Isometries (1970).- E. Calabi, Examples of Bernstein Problems for Some Nonlinear Equations (1970).- E. Calabi, Über singuläre symplektische Strukturen (1971).- E. Calabi, Complete Affine Hyperspheres I (1972).- E. Calabi, A Construction of Nonhomogeneous Einstein Metrics (1975).- E. Calabi, H. S. Wilf, On the Sequential and Random Selection of Subspaces Over a Finite Field (1977).- E. Calabi, Métriques kählériennes et fibrés holomorphes (1978).- E. Calabi, Isometric Families of Kähler Structures (1980).- E. Calabi, Géométrie différentielle affine des hypersurfaces (1981).- E. Calabi, Linear Systems of Real Quadratic Forms II (1982).- E. Calabi, Extremal Kähler Metrics (1982).- E. Calabi, Hypersurfaces with Maximal Affinely Invariant Area (1982).- E. Calabi, Extremal Kähler Metrics II (1985).- E. Calabi, Convex Affine Maximal Surfaces (1988).- E. Calabi, Affine Differential Geometry and Holomorphic Curves (1990).- E. Calabi, J. Cao Simple Closed Geodesics on Convex Surfaces (1992).- F. Beukers, J. A. C. Kolk and E. Calabi, Sums of Generalized Harmonic Series and Volumes (1993).- E. Calabi and H. Gluck, What are the Best Almost-Complex Structures on the 6-Sphere? (1993).- E. Calabi, Extremal Isosystolic Metrics for Compact Surfaces (1996).- E. Calabi, P. J. Olver, A. Tannenbaum, Affine Geometry, Curve Flows, and Invariant Numerical Approximations (1996).- J.-P. Bourguignon, E. Calabi, J. Eells, O. Garcia-Prada, M. Gromov, Where Does Geometry Go? A Research and Education Perspective (2001).- E. Calabi, X. Chen, The Space of Kähler Metrics II (2002).- Acknowledgements.
£123.49
Elsevier Science SemiRiemannian Geometry With Applications to
Book SynopsisTable of ContentsManifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.
£47.69
Oxford University Press Compact Manifolds with Special Holonomy
Book SynopsisThe book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kähler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkähler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of these manifolds were discovered by the author in 1993-5. This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.Trade ReviewThe book is written in a very clear and understandable way, with careful explanation of the main ideas and many remarks and comments, and it includes systematic suggestions for further reading ... It can be warmly recommended to mathematicians (in geometry and global analysis, in particular) as well as to physicists interested in string theory. * EMS *The first part is a very effective introduction to basic notions and results of modern differential geometry ... This book is highly recommended for people who are interested in the very recent developments of differential geometry and its relationships with present research in theoretical physics. * Zentralblatt MATH *
£159.75
Oxford University Press Geometry of Black Holes
Book SynopsisBlack holes present one of the most fascinating predictions of Einstein''s general theory of relativity. There is strong evidence of their existence through observation of active galactic nuclei, including the centre of our galaxy, observations of gravitational waves, and others.There exists a large scientific literature on black holes, including many excellent textbooks at various levels. However, most of these steer clear from the mathematical niceties needed to make the theory of black holes a mathematical theory. Those which maintain a high mathematical standard are either focused on specific topics, or skip many details. The objective of this book is to fill this gap and present a detailed, mathematically oriented, extended introduction to the subject.The book provides a wide background to the current research on all mathematical aspects of the geometry of black hole spacetimes.Trade ReviewWritten with a high standard of rigor and care, with very good treatments of many topics that are hard to find elsewhere. * Robert Wald, University of Chicago *Including some very interesting and unique material, the book is written in a manner that will be accessible for students, and provide a valuable resource for experts working in mathematical general relativity. * Greg Galloway, University of Miami *This text is an excellent research level monograph exploring the detailed and rich structure of black holes in mathematical physics. * Kymani Armstrong-Williams, Physics Book Reviews *Table of ContentsPART I GLOBAL LORENTZIAN GEOMETRY 1: Basic Notions 2: Elements of causality 3: Some applications PART II BLACK HOLES 4: An introduction to black holes 5: Further selected solutions 6: Extensions, conformal diagrams 7: Projection diagrams 8: Dynamical black holes
£37.99
Springer Riemannian Manifolds
Book SynopsisWhat Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.Trade Review"This book is very well writen, pleasant to read, with many good illustrations. It deals with the core of the subject, nothing more and nothing less. Simply a recommendation for anyone who wants to teach or learn about the Riemannian geometry."Nieuw Archief voor Wiskunde, September 2000Table of ContentsWhat Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.
£54.14
John Wiley & Sons Inc Conformal Differential Geometry and Its
Book SynopsisComprehensive coverage of the foundations, applications, recent developments, and future of conformal differential geometry. Conformal Differential Geometry and Its Generalizations systematically presents the foundations and manifestations of conformal differential geometry.Table of ContentsConformal and Pseudoconformal Spaces. Hypersurfaces in Conformal Spaces. Submanifolds in Conformal and Pseudoconformal Spaces. Conformal Structures on a Differentiable Manifold. The Four-Dimensional Conformal Structures. Geometry of the Grassmann Manifold. Manifolds Endowed with Almost Grassmann Structures. Bibliography. Symbols Frequently Used. Indexes.
£179.06
John Wiley & Sons Inc An Introduction to Integration and Measure Theory
Book SynopsisThis book describes integration and measure theory for readers interested in analysis, engineering, and economics. It gives a systematic account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes measure from the Lebesgue-Stieltjes integral.Table of ContentsLIMITATIONS OF THE RIEMANN INTEGRAL. Limits of Integrals and Integrability. Expectations in Probability Theory. RIEMANN-STIELTJES INTEGRALS. Riemann-Stieltjes Integrals: Introduction. Characterization of Riemann-Stieltjes Integrability. Continuous Linear Functionals on C[a,b]. Riemann-Stieltjes Integrals: Further Properties. LEBESGUE-STIELTJES INTEGRALS. The Extension of the Riemann-Stieltjes Integral. Lebesgue-Stieltjes Integrals. MEASURE THEORY. sigma-Algebras and Algebras of Sets. Measurable Functions. Measures. Lebesgue-Stieltjes Measures. THE ABSTRACT LEBESGUE INTEGRAL. The Integral Associated with a Measure Space. The Lebesgue Spaces and Norms. Absolutely Continuous Measures. Linear Functionals on the Lebesgue Spaces. Product Measures and Fubini's Theorem. Lebesgue Integration and Measures on R?n. Signed Measures and Complex Measures. Differentiation. Convergence of Sequences of Functions. Measures on Locally Compact Spaces. Hausdorff Measures and Dimension. Lorentz Spaces. Appendices. Indexes.
£165.56
Dover Publications Inc. Partial Differential Equations of Mathematical
Book Synopsis
£21.24
Dover Publications Inc. The Variational Theory of Geodesics Dover Books
Book SynopsisCompact, self-contained text by a noted theorist presents essentials of modern differential geometry and basic tools for study of Morse theory. Advanced treatment emphasizes Morse theory's analytical rather than topological aspects. 1967 edition.
£15.29
Dover Publications Inc. Theory of Lie Derivatives and Its Applications
Book SynopsisAdvanced treatment of topics in differential geometry, first published in 1957, was praised as "well written" by The American Mathematical Monthly and hailed as "undoubtedly a valuable addition to the literature."
£18.89
Cambridge University Press LMSST 11 Spacetime Singularities An Introduction London Mathematical Society Student Texts Series Number 11
Book SynopsisThis book is an elementary introduction to the geometrical methods and notions used in special and general relativity. Particular emphasis is placed on the ideas concerned with the structure of spacetime and those which play a role in the Penrose–Hawking singularity theorems.Trade Review"...fills a need to introduce the singularity theorems, their concepts and techniques to seniors or beginning graduate students." Physics in CanadaTable of ContentsPreface; 1. The geometry of Minkowski spacetime; 2. Some concepts from relativistic mechanics; 3. More general spacetimes: gravity; 4. The proof of Hawking's theorem; References; Index.
£44.99
Cambridge University Press An Introduction to Twistor Theory Second Edition 4 London Mathematical Society Student Texts Series Number 4
Book SynopsisThis book is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level. The choice of material presented has evolved from graduate lectures given in London and Oxford and the authors have aimed to retain the informal tone of those lectures. The book will provide graduate students with an introduction to the literature of twistor theory, presupposing some knowledge of special relativity and differential geometry. It would also be of use for a short course on space-time structure independently of twistor theory. The physicist could be introduced gently to some of the mathematics which has proved useful in these areas, and the mathematician could be shown where sheaf cohomology and complex manifold theory can be used in physics.Trade Review' … the book is recommended to anyone seeking to get acquainted with the area.' American Scientist' … a certain amount of preliminary knowledge is assumed of the reader ... but anyone who has these prerequisites and who is interested in twistor theory could hardly do better than to start with this book.' Contemporary Physics'In all, the book provides a pleasant starting point for the study of this fascinating subject.' Dr F. E. Burstall, Contemporary PhysicsTable of Contents1. Introduction; 2. Review of tensor algebra; 3. Lorentzian spinors at a point; 4. Spinor fields; 5. Compactified Minkowski space; 6. The geometry of null congruences; 7. The geometry of twistor space; 8. Solving the zero rest mass equations I; 9. Sheaf cohomology; 10. Solving the zero rest mass equations II; 11. The twisted photon and Yang–Mills constructions; 12. The non-linear graviton; 13. Penrose's quasi-local momentum; 14. Cohomological functionals; 15. Further developments and conclusion; Appendix: The GHP equations.
£127.30
Cambridge University Press An Introduction to Twistor Theory Second Edition 0004 London Mathematical Society Student Texts Series Number 4
Book SynopsisThis text is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level.Trade Review' … the book is recommended to anyone seeking to get acquainted with the area.' American Scientist' … a certain amount of preliminary knowledge is assumed of the reader ... but anyone who has these prerequisites and who is interested in twistor theory could hardly do better than to start with this book.' Contemporary Physics'In all, the book provides a pleasant starting point for the study of this fascinating subject.' Dr F. E. Burstall, Contemporary PhysicsTable of Contents1. Introduction; 2. Review of tensor algebra; 3. Lorentzian spinors at a point; 4. Spinor fields; 5. Compactified Minkowski space; 6. The geometry of null congruences; 7. The geometry of twistor space; 8. Solving the zero rest mass equations I; 9. Sheaf cohomology; 10. Solving the zero rest mass equations II; 11. The twisted photon and Yang–Mills constructions; 12. The non-linear graviton; 13. Penrose's quasi-local momentum; 14. Cohomological functionals; 15. Further developments and conclusion; Appendix: The GHP equations.
£41.79
Cambridge University Press A Primer of Algebraic DModules 33 London Mathematical Society Student Texts Series Number 33
Book SynopsisThe theory of D-modules is a rich area of study combining ideas from algebra and differential equations, and it has significant applications to diverse areas such as singularity theory and representation theory. This book introduces D-modules and their applications avoiding all unnecessary over-sophistication. It is aimed at beginning graduate students and the approach taken is algebraic, concentrating on the role of the Weyl algebra. Very few prerequisites are assumed, and the book is virtually self-contained. Exercises are included at the end of each chapter and the reader is given ample references to the more advanced literature. This is an excellent introduction to D-modules for all who are new to this area.Trade Review'I truly recommend this book, both for its mathematical content and for its light reading.' Bulletin of the London Mathematic Society'A readable account.' MathematikaTable of Contents1. The Weyl algebra; 2. Ideal structure of the Weyl algebra; 3. Rings of differential operators; 4. Jacobian conjectures; 5. Modules over the Weyl algebra; 6. Differential equations; 7. Graded and filtered modules; 8. Noetherian rings and modules; 9. Dimension and multiplicity; 10. Holonomic modules; 11. Characteristic varieties; 12. Tensor products; 13. External products; 14. Inverse image; 15. Embeddings; 16. Direct images; 17. Kashiwara's theorem; 18. Preservation of holonomy; 19. Stability of differential equations; 20. Automatic proof of identities.
£116.85
Cambridge University Press LMS 257 Intro to Noncomm Diff Geom London Mathematical Society Lecture Note Series Series Number 257
This thoroughly revised second edition includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Assuming only some familiarity with ordinary differential geometry and the theory of fibre bundles, this book is accessible to graduate students and newcomers to this field.
£74.73
Cambridge University Press Lectures on Khler Geometry 69 London Mathematical Society Student Texts Series Number 69
Book SynopsisKÃhler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. KÃhler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous KÃhler identities. The final part of the text studies several aspects of compact KÃhler manifolds: the Calabi conjecture, WeitzenbÃck techniques, CalabiâYau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.Trade Review"A concise and well-written modern introduction to the subject." Tatyana E. Foth, Mathematical ReviewsTable of ContentsIntroduction; Part I. Basics on Differential Geometry: 1. Smooth manifolds; 2. Tensor fields on smooth manifolds; 3. The exterior derivative; 4. Principal and vector bundles; 5. Connections; 6. Riemannian manifolds; Part II. Complex and Hermitian Geometry: 7. Complex structures and holomorphic maps; 8. Holomorphic forms and vector fields; 9. Complex and holomorphic vector bundles; 10. Hermitian bundles; 11. Hermitian and Kähler metrics; 12. The curvature tensor of Kähler manifolds; 13. Examples of Kähler metrics; 14. Natural operators on Riemannian and Kähler manifolds; 15. Hodge and Dolbeault theory; Part III. Topics on Compact Kähler Manifolds: 16. Chern classes; 17. The Ricci form of Kähler manifolds; 18. The Calabi–Yau theorem; 19. Kähler–Einstein metrics; 20. Weitzenböck techniques; 21. The Hirzebruch–Riemann–Roch formula; 22. Further vanishing results; 23. Ricci–flat Kähler metrics; 24. Explicit examples of Calabi–Yau manifolds; Bibliography; Index.
£40.99
Cambridge University Press Elementary Differential Geometry
Book SynopsisThis easy-to-read, generously illustrated textbook is an elementary introduction to differential geometry with emphasis on geometric results, preparing students for more advanced study. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and self-study.Trade Review'The book under review presents a detailed and pedagogically excellent study about differential geometry of curves and surfaces by introducing modern concepts and techniques so that it can serve as a transition book between classical differential geometry and contemporary theory of manifolds. the concepts are discussed through historical problems as well as motivating examples and applications. Moreover, constructive examples are given in such a way that the reader can easily develop some understanding for extensions, generalizations and adaptations of classical differential geometry to global differential geometry.' Zentralblatt MATHTable of ContentsPreface; Notation; 1. Euclidean geometry; 2. Curve theory; 3. Classical surface theory; 4. The inner geometry of surfaces; 5. Geometry and analysis; 6. Geometry and topology; 7. Hints for solutions to (most) exercises; Formulary; List of symbols; References; Index.
£49.99
£11.77
Princeton University Press Seminar on the AtiyahSinger Index Theorem
Book SynopsisThe description for this book, Seminar on Atiyah-Singer Index Theorem. (AM-57), will be forthcoming.Table of Contents*Frontmatter, pg. i*CONTENTS, pg. v*PREFACE, pg. ix*CHAPTER I. STATEMENT OF THE THEOREM OUTLINE OF THE PROOF, pg. 1*CHAPTER II. REVIEW OF K-THEORY, pg. 13*CHAPTER III. THE TOPOLOGICAL INDEX OF AN OPERATOR ASSOCIATED TO A G-STRUCTURE, pg. 27*CHAPTER IV. DIFFERENTIAL OPERATORS ON VECTOR BUNDLES, pg. 51*CHAPTER V. ANALYTICAL INDICES OF SOME CONCRETE OPERATORS, pg. 95*CHAPTER VI. REVIEW OF FUNCTIONAL ANALYSIS, pg. 107*CHAPTER VII. FREDHDIM OPERATORS, pg. 119*CHAPTER VIII. CHAINS OP HILBERTIAN SPACES, pg. 125*CHAPTER IX. THE DISCRETE SOBOLEV CHAIN OF A VECTOR BUNDLE, pg. 147*CHAPTER X. THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE, pg. 155*CHAPTER XI. THE SEELEY ALGEBRA, pg. 175*CHAPTER XII. HOMOTOPY INVARIANCE OF THE INDEX, pg. 185*CHAPTER XIII. WHITNEY SUMS, pg. 191*CHAPTER XIV. TENSOR PRODUCTS, pg. 197*CHAPTER XV. DEFINITION OF ia AND it ON K(M), pg. 215*CHAPTER XVI. CONSTRUCTION OF Intk, pg. 235*CHAPTER XVII. COBORDISM INVARIANCE OP THE ANALYTICAL INDEX, pg. 285*CHAPTER XVIII. BORDISM GROUPS OF BUNDLES, pg. 303*CHAPTER XIX. THE INDEX THEOREM: APPLICATIONS, pg. 313*APPENDIX I. THE INDEX THEOREM FOR MANIFOLDS WITH BOUNDARY, pg. 337*APPENDIX II. NON-STABLE CHARACTERISTIC CLASSES AND THE TOPOLOGICAL INDEX OP CLASSICAL ELLIPTIC OPERATORS, pg. 353*Backmatter, pg. 368
£87.20
Princeton University Press Characteristic Classes
Book SynopsisTrade Review"John Milnor, Winner of the 2011 Abel Prize from the Norwegian Academy of Science and Letters""John Willard Milnor, Winner of the 2011 Leroy P. Steele Prize for Lifetime Achievement, American Mathematical Society"Table of Contents*Frontmatter, pg. i*Preface, pg. v*Contents, pg. vii* 1. Smooth Manifolds, pg. 1* 2. Vector Bundles, pg. 13* 3. Constructing New Vector Bundles Out of Old, pg. 25* 4. Stiefel-Whitney Classes, pg. 37* 5. Grassmann Manifolds and Universal Bundles, pg. 55* 6. A Cell Structure for Grassmann Manifolds, pg. 73* 7. The Cohomology Ring H*(Gn; Z/2), pg. 83* 8. Existence of Stiefel-Whitney Classes, pg. 89* 9. Oriented Bundles and the Euler Class, pg. 95* 10. The Thom Isomorphism Theorem, pg. 105* 11. Computations in a Smooth Manifold, pg. 115* 12. Obstructions, pg. 139* 13. Complex Vector Bundles and Complex Manifolds, pg. 149* 14. Chern Classes, pg. 155* 15. Pontrjagin Classes, pg. 173* 16. Chern Numbers and Pontrjagin Numbers, pg. 183* 17. The Oriented Cobordism Ring OMEGA*, pg. 199* 18. Thom Spaces and Transversality, pg. 205* 19. Multiplicative Sequences and the Signature Theorem, pg. 219* 20. Combinatorial Pontrjagin Classes, pg. 231*Epilogue, pg. 249*Appendix A: Singular Homology and Cohomology, pg. 257*Appendix B: Bernoulli Numbers, pg. 281*Appendix C: Connections, Curvature, and Characteristic Classes, pg. 289*Bibliography, pg. 315*Index, pg. 325
£87.20
Princeton University Press The Decomposition of Global Conformal Invariants
Book SynopsisAddresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies.Table of Contents*FrontMatter, pg. i*Contents, pg. v*Acknowledgments, pg. vii*1. Introduction, pg. 1*2. An Iterative Decomposition of Global Conformal Invariants: The First Step, pg. 19*3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition, pg. 71*4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition, pg. 135*5. The Inductive Step of the Fundamental Proposition: The Simpler Cases, pg. 211*6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I, pg. 297*7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II, pg. 361*A. Appendix, pg. 403*Bibliography, pg. 443*Index of Authors and Terms, pg. 447*Index of Symbols, pg. 449
£160.00
Princeton University Press HypoAnalytic Structures
Book SynopsisIn 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 geometriTable of ContentsPrefaceIFormally and Locally Integrable Structures. Basic Definitions3I.1Involutive systems of linear PDE defined by complex vector fields. Formally and locally integrable structures5I.2The characteristic set. Partial classification of formally integrable structures11I.3Strongly noncharacteristic, totally real, and maximally real submanifolds16I.4Noncharacteristic and totally characteristic submanifolds23I.5Local representations27I.6The associated differential complex32I.7Local representations in locally integrable structures39I.8The Levi form in a formally integrable structure46I.9The Levi form in a locally integrable structure49I.10Characteristics in real and in analytic structures56I.11Orbits and leaves. Involutive structures of finite type63I.12A model case: Tube structures68IILocal Approximation and Representation in Locally Integrable Structures73II.1The coarse local embedding76II.2The approximation formula81II.3Consequences and generalizations86II.4Analytic vectors94II.5Local structure of distribution solutions and of L-closed currents100II.6The approximate Poincare lemma104II.7Approximation and local structure of solutions based on the fine local embedding108II.8Unique continuation of solutions115IIIHypo-Analytic Structures. Hypocomplex Manifolds120III.1Hypo-analytic structures121III.2Properties of hypo-analytic functions128III.3Submanifolds compatible with the hypo-analytic structure130III.4Unique continuation of solutions in a hypo-analytic manifold137III.5Hypocomplex manifolds. Basic properties145III.6Two-dimensional hypocomplex manifolds152Appendix to Section III.6: Some lemmas about first-order differential operators159III.7A class of hypocomplex CR manifolds162IVIntegrable Formal Structures. Normal Forms167IV.1Integrable formal structures168IV.2Hormander numbers, multiplicities, weights. Normal forms174IV.3Lemmas about weights and vector fields178IV.4Existence of basic vector fields of weight - 1185IV.5Existence of normal forms. Pluriharmonic free normal forms. Rigid structures191IV.6Leading parts198VInvolutive Structures with Boundary201V.1Involutive structures with boundary202V.2The associated differential complex. The boundary complex209V.3Locally integrable structures with boundary. The Mayer-Vietoris sequence219V.4Approximation of classical solutions in locally integrable structures with boundary226V.5Distribution solutions in a manifold with totally characteristic boundary228V.6Distribution solutions in a manifold with noncharacteristic boundary235V.7Example: Domains in complex space246VILocal Integrability and Local Solvability in Elliptic Structures252VI.1The Bochner-Martinelli formulas253VI.2Homotopy formulas for [actual symbol not reproducible] in convex and bounded domains258VI.3Estimating the sup norms of the homotopy operators264VI.4Holder estimates for the homotopy operators in concentric balls269VI.5The Newlander-Nirenberg theorem281VI.6End of the proof of the Newlander-Nirenberg theorem287VI.7Local integrability and local solvability of elliptic structures. Levi flat structures291VI.8Partial local group structures297VI.9Involutive structures with transverse group action. Rigid structures. Tube structures303VIIExamples of Nonintegrability and of Nonsolvability312VII.1Mizohata structures314VII.2Nonsolvability and nonintegrability when the signature of the Levi form is |n - 2|319VII.3Mizohata structures on two-dimensional manifolds324VII.4Nonintegrability and nonsolvability when the cotangent structure bundle has rank one330VII.5Nonintegrability and nonsolvability in Lewy structures. The three-dimensional case337VII.6Nonintegrability in Lewy structures. The higher-dimensional case343VII.7Example of a CR structure that is not locally integrable but is locally integrable on one side348VIIINecessary Conditions for the Vanishing of the Cohomology. Local Solvability of a Single Vector Field352VIII.1Preliminary necessary conditions for exactness354VIII.2Exactness of top-degree forms358VIII.3A necessary condition for local exactness based on the Levi form364VIII.4A result about structures whose characteristic set has rank at most equal to one367VIII.5Proof of Theorem VIII.4.1373VIII.6Applications of Theorem VII
£70.40
Birkhauser Boston Inc Riemannian Geometry
Book SynopsisRiemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics.Trade Review"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry." -Publicationes Mathematicae "This is a very nice introduction to global Riemannian geometry, which leads the reader quickly to the heart of the topic. Nevertheless, classical results are also discussed on many occasions, and almost 60 pages are devoted to exercises." -Newsletter of the EMS "In the reviewer's opinion, this is a superb book which makes learning a real pleasure." -Revue Romaine de Mathematiques Pures et Appliquees "This mainstream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises." -Monatshefte F. MathematikTable of ContentsPreface to the 1st edition * Preface to the 2nd edition * Preface to the English edition * How to use this book * 0. Differentiable Manifolds * 1. Riemannian Metrics * 2. Affine Connections; Riemannian Connections * 3. Geodesics; Convex Neighborhoods * 4. Curvature * 5. Jacobi Fields * 6. Isometric Immersions * 7. Complete Manifolds; Hopf-Rinow and Hadamard Theorems * 8. Spaces of Constant Curvature * 9. Variations of Energy * 10. The Rauch Comparison Theorem * 11. The Morse Index Theorem * 12. The Fundamental Group of Manifolds of Negative Curvature * 13. The Sphere Theorem * References * Index
£35.99
Birkhauser Boston Representation Theory and Automorphic Forms
Book SynopsisThis volume uses a unified approach to representation theory and automorphic forms.Table of ContentsIntroduction.- Ramakrishnan, D.: Irreducibility and Cuspidality.-Ikeda, T.: On Liftings of Holomorphic Modular Forms.-Kobayashi, T.: Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs.-Miller, S., Schmid, W.: The Rankin--Selberg Method for Automorphic Distributions.- Shahidi, F.: Langlands Functoriality Conjecture and Number Theory.- Yoshikawa, K.: Discriminant of certain K3 surfaces.- References.- Index.
£104.49
MP-AMM American Mathematical Fundamental Groups of Compact Kahler Manifolds
Book SynopsisAn exposition of what is known about the fundamental groups of compact Kahler manifolds. It collects together various results obtained over the years which aim to characterise those infinite groups which can arise as fundamental groups of compact Kahler manifolds. Most of these results are negative ones, saying which groups do not arise.Table of ContentsIntroduction Fibering Kahler manifolds and Kahler groups The de Rham fundamental group $L^2$-cohomology of Kahler groups Existence theorems for harmonic maps Applications of harmonic maps Non-Abelian Hodge theory Positive results for infinite groups Pro group theory (Appendix A) A glossary of Hodge theory (Appendix B) Bibliography Index.
£96.30
MP-AMM American Mathematical The Convenient Setting of Global Analysis
Book SynopsisFocuses on differential calculus in infinite dimensions and those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. This work discusses the existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions and differential forms.Table of ContentsIntroduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.
£101.70
MP-AMM American Mathematical Global Analysis
Book SynopsisPresents an introduction to differential geometry through differential forms, emphasizing their applications in various areas of mathematics and physics. This work focuses on Stokes' theorem, the classical integral formulas and their applications to harmonic functions and topology.Table of ContentsElements of multilinear algebra Differential forms in ${\mathbb{R}}^n$ Vector analysis on manifolds Pfaffian systems Curves and surfaces in Euclidean 3-space Lie groups and homogeneous spaces Symplectic geometry and mechanics Elements of statistical mechanics and thermodynamics Elements of electrodynamics Bibliography Symbols Index.
£79.20
MP-AMM American Mathematical Nonlinear Dynamics and Evolution Equations
Book SynopsisReflects a broad spectrum of research activities on the theory and applications of nonlinear dynamics and evolution equations. This book covers major areas of dynamical systems, ordinary, functional and partial differential equations, and applications of such equations in the mathematical modelling of various biological and physical phenomena.Table of ContentsDisease spread in metapopulations by J. Arino and P. van den Driessche On some nonlocal evolution equations arising in materials science by P. W. Bates Invariant tori for Hamiltonian PDE by W. Craig Stable and not too unstable solutions on $R^n$ for small diffusion by N. Dancer Some recent results on diffusive predator-prey models in spatially heterogeneous environment by Y. Du and J. Shi Delayed non-local diffusive systems in biological invasion and disease spread by S. A. Gourley and J. Wu Asymptotic behavior for systems comparable to quasimonotone systems by J. Jiang $C^1$-smoothness of center manifolds for differential equations with state-dependent delay by T. Krisztin Normal forms for germs of analytic families of planar vector fields unfolding a generic saddle-node or resonant saddle by C. Rousseau Generic properties of symplectic diffeomorphisms by R. Saghin and Z. Xia Mathematical aspects of modelling tumour angiogenesis by B. D. Sleeman Interpretation of the generalized asymmetric May-Leonard model of three species competition as a food web in a chemostat by G. S. K. Wolkowicz On exact Poisson structures by Y. Yi and X. Zhang.
£99.00
Springer London Geodesic and Horocyclic Trajectories
Book SynopsisGeodesic and Horocyclic Trajectories presents an introduction to the topological dynamics of two classical flows associated with surfaces of curvature −1, namely the geodesic and horocycle flows.Table of ContentsDynamics of Fuchsian groups.- Examples of Fuchsian Groups.- Topological dynamics of the geodesic flow.- Schottky groups.- Topological dynamics.- The Lorentzian point of view.- Trajectories and Diophantine approximations.
£52.24
Tarquin Publications Geodesic Domes
Book Synopsis
£9.49
Deductive Press Introduction to Arithmetic Groups
£14.53
Deductive Press Introduction to Arithmetic Groups
£20.38
Legare Street Press Théorie Mathématique De La Lumière Ii.
Book Synopsis
£19.90
Taylor & Francis Ltd Classical and Discrete Differential Geometry
Book SynopsisThis book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks.With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geomet
£48.75
Cambridge University Press The Geometry of Celestial Mechanics 83 London Mathematical Society Student Texts Series Number 83
Book SynopsisCelestial mechanics is the branch of mathematical astronomy devoted to studying the motions of celestial bodies subject to the Newtonian law of gravitation. This mathematical introductory textbook reveals that even the most basic question in celestial mechanics, the Kepler problem, leads to a cornucopia of geometric concepts: conformal and projective transformations, spherical and hyperbolic geometry, notions of curvature, and the topology of geodesic flows. For advanced undergraduate and beginning graduate students, this book explores the geometric concepts underlying celestial mechanics and is an ideal companion for introductory courses. The focus on the history of geometric ideas makes it perfect supplementary reading for students in elementary geometry and topology. Numerous exercises, historical notes and an extensive bibliography provide all the contextual information required to gain a solid grounding in celestial mechanics.Trade Review'The Geometry of Celestial Mechanics offers a fresh look at one of the most celebrated topics of mathematics … I would gladly recommend this book …' Anil Venkatesh, Mathematical Association of America Reviews'Because much of the geometric theory, the many historical notes, and the exercises in the book are not found in other contemporary books on celestial mechanics, the book makes a great addition to the library of anyone with an interest in celestial mechanics.' Lennard Bakker, Zentralblatt MATH'The book fulfills the authors quest, as stated in the preface, 'for students to experience differential geometry and topology 'in action' (in the historical context of celestial mechanics) rather than as abstractions in traditional courses on the two subjects.' Lennard F. Bakker, Mathematical ReviewsTable of ContentsPreface; 1. The central force problem; 2. Conic sections; 3. The Kepler problem; 4. The dynamics of the Kepler problem; 5. The two-body problem; 6. The n-body problem; 7. The three-body problem; 8. The differential geometry of the Kepler problem; 9. Hamiltonian mechanics; 10. The topology of the Kepler problem; Bibliography; Index.
£36.99
Cambridge University Press Constrained Willmore Surfaces
Book SynopsisFrom Bäcklund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Bäcklund transformations and Darboux transformations, and proves that all these transformations preserve the existence of a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed compTable of ContentsIntroduction; 1. A bundle approach to conformal surfaces in space-forms; 2. The mean curvature sphere congruence; 3. Surfaces under change of flat metric connection; 4. Willmore surfaces; 5. The Euler–Lagrange constrained Willmore surface equation; 6. Transformations of generalized harmonic bundles and constrained Willmore surfaces; 7. Constrained Willmore surfaces with a conserved quantity; 8. Constrained Willmore surfaces and the isothermic surface condition; 9. The special case of surfaces in 4-space; Appendix A. Hopf differential and umbilics; Appendix B. Twisted vs. untwisted Bäcklund transformation parameters; References; Index.
£55.09
Cambridge University Press Differential Geometry in the Large
Book SynopsisThe 2019 ''Australian-German Workshop on Differential Geometry in the Large'' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kähler and Sasaki geometry.Trade Review'The high-quality surveys and original work in this book give a convenient path to understand some recent exciting developments in global Differential Geometry and Geometric Analysis. This should be of great value to graduate students entering the field, as well as to more experienced researchers looking for an updated perspective on a wide range of topics, ranging from nonnegative curvature and Alexandrov spaces to geometric flows and equivariant geometry.' Renato G. Bettiol, Lehman College, The City University of New York'The volume includes important additions to the literature including new results, new proofs of previous results, and simplified expositions, and also an excellent collection of surveys on recent activity. It is well written and offers a generous overview and invitation to a variety of modern, active topics in differential geometry.' Christopher Seaton, MAA ReviewsTable of ContentsIntroduction Owen Dearricott, Wilderich Tuschmann, Yuri Nikolayevsky, Thomas Leistner and Diarmuid Crowley; Part I. Geometric Evolution Equations and Curvature Flow: 1. Real geometric invariant theory Christoph Böhm and Ramiro A. Lafuente; 2. Convex ancient solutions to mean curvature flow Theodora Bourni, Mat Langford and Giuseppe Tinaglia; 3. Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in Minkowski space and the cross curvature flow Paul Bryan, Mohammad N. Ivaki and Julian Scheuer; 4. A mean curvature flow for conformally compact manifolds A. Rod Gover and Valentina-Mira Wheeler; 5. A survey on the Ricci flow on singular spaces Klaus Kröncke and Boris Vertman; Part II. Structures on Manifolds and Mathematical Physics: 6. Some open problems in Sasaki geometry Charles P. Boyer, Hongnian Huang, Eveline Legendre and Christina W. Tønnesen-Friedman; 7. The prescribed Ricci curvature problem for homogeneous metrics Timothy Buttsworth and Artem Pulemotov; 8. Singular Yamabe and Obata problems A. Rod Gover and Andrew K. Waldron; 9. Einstein metrics, harmonic forms and conformally Kähler geometry Claude LeBrun; 10. Construction of the supersymmetric path integral: a survey Matthias Ludewig; 11. Tight models of de-Rham algebras of highly connected manifolds Lorenz Schwachhöfer; Part III. Recent Developments in Non-Negative Sectional Curvature: 12. Fake lens spaces and non-negative sectional curvature Sebastian Goette, Martin Kerin and Krishnan Shankar; 13. Collapsed three-dimensional Alexandrov spaces: a brief survey Fernando Galaz-García, Luis Guijarro and Jesús Núñez-Zimbrón; 14. Pseudo-angle systems and the simplicial Gauss–Bonnet–Chern theorem Stephan Klaus; 15. Aspects and examples on quantitative stratification with lower curvature bounds Nan Li; 16. Universal covers of Ricci limit and RCD spaces Jiayin Pan and Guofang Wei; 17. Local and global homogeneity for manifolds of positive curvature Joseph A. Wolf.
£43.69
John Wiley & Sons Inc Introduction to Differential Geometry with Tensor
Book SynopsisINTRODUCTION TO DIFFERENTIAL GEOMETRY WITH TENSOR APPLICATIONS This is the only volume of its kind to explain, in precise and easy-to-understand language, the fundamentals of tensors and their applications in differential geometry and analytical mechanics with examples for practical applications and questions for use in a course setting. Introduction to Differential Geometry with Tensor Applications discusses the theory of tensors, curves and surfaces and their applications in Newtonian mechanics. Since tensor analysis deals with entities and properties that are independent of the choice of reference frames, it forms an ideal tool for the study of differential geometry and also of classical and celestial mechanics. This book provides a profound introduction to the basic theory of differential geometry: curves and surfaces and analytical mechanics with tensor applications. The author has tried to keep the treatment of the advanced material as lucid and comprehensive as possible, mainlyTable of ContentsPreface xv About the Book xvii Introduction 1 Part I: Tensor Theory 7 1 Preliminaries 9 1.1 Introduction 9 1.2 Systems of Different Orders 9 1.3 Summation Convention Certain Index 10 1.3.1 Dummy Index 11 1.3.2 Free Index 11 1.4 Kronecker Symbols 11 1.5 Linear Equations 14 1.6 Results on Matrices and Determinants of Systems 15 1.7 Differentiation of a Determinant 18 1.8 Examples 19 1.9 Exercises 23 2 Tensor Algebra 25 2.1 Introduction 25 2.2 Scope of Tensor Analysis 25 2.2.1 n-Dimensional Space 26 2.3 Transformation of Coordinates in S n 27 2.3.1 Properties of Admissible Transformation of Coordinates 30 2.4 Transformation by Invariance 31 2.5 Transformation by Covariant Tensor and Contravariant Tensor 32 2.6 The Tensor Concept: Contravariant and Covariant Tensors 34 2.6.1 Covariant Tensors 34 2.6.2 Contravariant Vectors 35 2.6.3 Tensor of Higher Order 40 2.6.3.1 Contravariant Tensors of Order Two 40 2.6.3.2 Covariant Tensor of Order Two 41 2.6.3.3 Mixed Tensors of Order Two 42 2.7 Algebra of Tensors 43 2.7.1 Equality of Two Tensors of Same Type 45 2.8 Symmetric and Skew-Symmetric Tensors 45 2.8.1 Symmetric Tensors 45 2.8.2 Skew-Symmetric Tensors 46 2.9 Outer Multiplication and Contraction 51 2.9.1 Outer Multiplication 51 2.9.2 Contraction of a Tensor 53 2.9.3 Inner Product of Two Tensors 54 2.10 Quotient Law of Tensors 56 2.11 Reciprocal Tensor of a Tensor 58 2.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors 60 2.12.1 Relative Tensors 60 2.12.2 Cartesian Tensors 63 2.12.3 Affine Tensors 63 2.12.4 Isotropic Tensor 64 2.12.5 Pseudo-Tensors 64 2.13 Examples 65 2.14 Exercises 71 3 Riemannian Metric 73 3.1 Introduction 73 3.2 The Metric Tensor 74 3.3 Conjugate Tensor 75 3.4 Associated Tensors 77 3.5 Length of a Vector 84 3.5.1 Length of Vector 84 3.5.2 Unit Vector 85 3.5.3 Null Vector 86 3.6 Angle Between Two Vectors 86 3.6.1 Orthogonality of Two Vectors 87 3.7 Hypersurface 88 3.8 Angle Between Two Coordinate Hypersurfaces 89 3.9 Exercises 95 4 Tensor Calculus 97 4.1 Introduction 97 4.2 Christoffel Symbols 97 4.2.1 Properties of Christoffel Symbols 98 4.3 Transformation of Christoffel Symbols 110 4.3.1 Law of Transformation of Christoffel Symbols of 1st Kind 110 4.3.2 Law of Transformation of Christoffel Symbols of 2nd Kind 111 4.4 Covariant Differentiation of Tensor 113 4.4.1 Covariant Derivative of Covariant Tensor 114 4.4.2 Covariant Derivative of Contravariant Tensor 115 4.4.3 Covariant Derivative of Tensors of Type (0,2) 116 4.4.4 Covariant Derivative of Tensors of Type (2,0) 118 4.4.5 Covariant Derivative of Mixed Tensor of Type (s, r) 120 4.4.6 Covariant Derivatives of Fundamental Tensors and the Kronecker Delta 120 4.4.7 Formulas for Covariant Differentiation 122 4.4.8 Covariant Differentiation of Relative Tensors 123 4.5 Gradient, Divergence, and Curl 129 4.5.1 Gradient 130 4.5.2 Divergence 130 4.5.2.1 Divergence of a Mixed Tensor (1,1) 132 4.5.3 Laplacian of an Invariant 136 4.5.4 Curl of a Covariant Vector 137 4.6 Exercises 141 5 Riemannian Geometry 143 5.1 Introduction 143 5.2 Riemannian-Christoffel Tensor 143 5.3 Properties of Riemann-Christoffel Tensors 150 5.3.1 Space of Constant Curvature 158 5.4 Ricci Tensor, Bianchi Identities, Einstein Tensors 159 5.4.1 Ricci Tensor 159 5.4.2 Bianchi Identity 160 5.4.3 Einstein Tensor 166 5.5 Einstein Space 170 5.6 Riemannian and Euclidean Spaces 171 5.6.1 Riemannian Spaces 171 5.6.2 Euclidean Spaces 174 5.7 Exercises 175 6 The e-Systems and the Generalized Kronecker Deltas 177 6.1 Introduction 177 6.2 e-Systems 177 6.3 Generalized Kronecker Delta 181 6.4 Contraction of δijk αβγ 183 6.5 Application of e-Systems to Determinants and Tensor Characters of Generalized Kronecker Deltas 185 6.5.1 Curl of Covariant Vector 189 6.5.2 Vector Product of Two Covariant Vectors 190 6.6 Exercises 192 Part II: Differential Geometry 193 7 Curvilinear Coordinates in Space 195 7.1 Introduction 195 7.2 Length of Arc 195 7.3 Curvilinear Coordinates in E 3 200 7.3.1 Coordinate Surfaces 201 7.3.2 Coordinate Curves 202 7.3.3 Line Element 205 7.3.4 Length of a Vector 206 7.3.5 Angle Between Two Vectors 207 7.4 Reciprocal Base Systems 210 7.5 Partial Derivative 216 7.6 Exercises 219 8 Curves in Space 221 8.1 Introduction 221 8.2 Intrinsic Differentiation 221 8.3 Parallel Vector Fields 226 8.4 Geometry of Space Curves 228 8.4.1 Plane 231 8.5 Serret-Frenet Formula 233 8.5.1 Bertrand Curves 235 8.6 Equations of a Straight Line 252 8.7 Helix 254 8.7.1 Cylindrical Helix 256 8.7.2 Circular Helix 258 8.8 Exercises 262 9 Intrinsic Geometry of Surfaces 265 9.1 Introduction 265 9.2 Curvilinear Coordinates on a Surface 265 9.3 Intrinsic Geometry: First Fundamental Quadratic Form 267 9.3.1 Contravariant Metric Tensor 270 9.4 Angle Between Two Intersecting Curves on a Surface 272 9.4.1 Pictorial Interpretation 274 9.5 Geodesic in R n 277 9.6 Geodesic Coordinates 289 9.7 Parallel Vectors on a Surface 291 9.8 Isometric Surface 292 9.8.1 Developable 293 9.9 The Riemannian–Christoffel Tensor and Gaussian Curvature 294 9.9.1 Einstein Curvature 296 9.10 The Geodesic Curvature 308 9.11 Exercises 319 10 Surfaces in Space 321 10.1 Introduction 321 10.2 The Tangent Vector 321 10.3 The Normal Line to the Surface 324 10.4 Tensor Derivatives 329 10.5 Second Fundamental Form of a Surface 332 10.5.1 Equivalence of Definition of Tensor b αβ 333 10.6 The Integrability Condition 334 10.7 Formulas of Weingarten 337 10.7.1 Third Fundamental Form 338 10.8 Equations of Gauss and Codazzi 339 10.9 Mean and Total Curvatures of a Surface 341 10.10 Exercises 347 11 Curves on a Surface 349 11.1 Introduction 349 11.2 Curve on a Surface: Theorem of Meusnier 350 11.2.1 Theorem of Meusnier 353 11.3 The Principal Curvatures of a Surface 358 11.3.1 Umbillic Point 360 11.3.2 Lines of Curvature 361 11.3.3 Asymptotic Lines 362 11.4 Rodrigue’s Formula 376 11.5 Exercises 379 12 Curvature of Surface 381 12.1 Introduction 381 12.2 Surface of Positive and Negative Curvatures 381 12.3 Parallel Surfaces 383 12.3.1 Computation of aαβ and b αβ 383 12.4 The Gauss-Bonnet Theorem 387 12.5 The n-Dimensional Manifolds 391 12.6 Hypersurfaces 394 12.7 Exercises 395 Part III: Analytical Mechanics 397 13 Classical Mechanics 399 13.1 Introduction 399 13.2 Newtonian Laws of Motion 399 13.3 Equations of Motion of Particles 401 13.4 Conservative Force Field 403 13.5 Lagrangean Equations of Motion 405 13.6 Applications of Lagrangean Equations 411 13.7 Himilton’s Principle 423 13.8 Principle of Least Action 427 13.9 Generalized Coordinates 430 13.10 Lagrangean Equations in Generalized Coordinates 432 13.11 Divergence Theorem, Green’s Theorem, Laplacian Operator, and Stoke’s Theorem in Tensor Notation 438 13.12 Hamilton’s Canonical Equations 442 13.12.1 Generalized Momenta 443 13.13 Exercises 444 14 Newtonian Law of Gravitations 447 14.1 Introduction 447 14.2 Newtonian Laws of Gravitation 447 14.3 Theorem of Gauss 451 14.4 Poisson’s Equation 453 14.5 Solution of Poisson’s Equation 454 14.6 The Problem of Two Bodies 456 14.7 The Problem of Three Bodies 462 14.8 Exercises 467 Appendix A: Answers to Even-Numbered Exercises 469 References 473 Index 475
£153.90
Springer Science+Business Media An Introduction to Manifolds
Book SynopsisManifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.Trade ReviewFrom the reviews of the second edition:“This book could be called a prequel to the book ‘Differential forms in algebraic topology’ by R. Bott and the author. Assuming only basic background in analysis and algebra, the book offers a rather gentle introduction to smooth manifolds and differential forms offering the necessary background to understand and compute deRham cohomology. … The text also contains many exercises … for the ambitious reader.” (A. Cap, Monatshefte für Mathematik, Vol. 161 (3), October, 2010)Table of ContentsPreface to the Second Edition.- Preface to the First Edition.-Chapter 1. Eudlidean Spaces. 1. Smooth Functions on a Euclidean Space.- 2. Tangent Vectors in R(N) as Derivativations.- 3. The Exterior Algebra of Multicovectors.- 4. Differential Forms on R(N).- Chapter 2. Manifolds.- 5. Manifolds.- 6. Smooth Maps on a Manifold.- 7. Quotients.- Chapter 3. The Tangent Space.- 8. The Tangent Space.- 9. Submanifolds.- 10. Categories and Functors.- 11. The Rank of a Smooth Map.- 12. The Tangent Bundle.- 13. Bump Functions and Partitions of Unity.- 14. Vector Fields.-Chapter 4. Lie Groups and Lie Algebras.- 15. Lie Groups.- 16. Lie Algebras.- Chapter 5. Differential Forms.- 17. Differential 1-Forms.- 18. Differential k-Forms.- 19. The Exterior Derivative.- 20. The Lie Derivative and Interior Multiplication.- Chapter 6. Integration.- 21. Orientations.- 22. Manifolds with Boundary.- 23. Integration on Manifolds.- Chapter 7. De Rham Theory.- 24. De Rham Cohomology.- 25. The Long Exact Sequence in Cohomology.- 26. The Mayer –Vietoris Sequence.- 27. Homotopy Invariance.- 28. Computation of de Rham Cohomology.- 29. Proof of Homotopy Invariance.-Appendices.- A. Point-Set Topology.- B. The Inverse Function Theorem on R(N) and Related Results.- C. Existence of a Partition of Unity in General.- D. Linear Algebra.- E. Quaternions and the Symplectic Group.- Solutions to Selected Exercises.- Hints and Solutions to Selected End-of-Section Problems.- List of Symbols.- References.- Index.
£42.74
Springer-Verlag New York Inc. Introduction to Smooth Manifolds
Book SynopsisThis book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theTrade ReviewFrom the reviews of the second edition:“It starts off with five chapters covering basics on smooth manifolds up to submersions, immersions, embeddings, and of course submanifolds. … the book under review is laden with excellent exercises that significantly further the reader’s understanding of the material, and Lee takes great pains to motivate everything well all the way through … . a fine graduate-level text for differential geometers as well as people like me, fellow travelers who always wish they knew more about such a beautiful subject.” (Michael Berg, MAA Reviews, October, 2012)Table of ContentsPreface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves and Flows.- 10 Vector Bundles.- 11 The Cotangent Bundle.- 12 Tensors.- 13 Riemannian Metrics.- 14 Differential Forms.- 15 Orientations.- 16 Integration on Manifolds.- 17 De Rham Cohomology.- 18 The de Rham Theorem.- 19 Distributions and Foliations.- 20 The Exponential Map.- 21 Quotient Manifolds.- 22 Symplectic Manifolds.- Appendix A: Review of Topology.- Appendix B: Review of Linear Algebra.- Appendix C: Review of Calculus.- Appendix D: Review of Differential Equations.- References.- Notation Index.- Subject Index
£56.99
Read Books Plane And Solid Analytic Geometry
£28.49
Springer London Morse Theory and Floer Homology
Book SynopsisIt defines the Morse complex and the Morse homology, and develops some of their applications.Morse homology also serves a simple model for Floer homology, which is covered in the second part.Trade ReviewFrom the book reviews:“The present book is an excellent, detailed and self-contained introduction to Morse theory and Floer homology which makes both topics easily accessible to graduate or even advanced undergraduate students.” (Sonja Hohloch, Mathematical Reviews, August, 2014)“Morse Theory and Floer Homology is a relatively high-level introduction to, and in fact a full account of, the extremely elegant and properly celebrated solution to the Arnol’d problem by the prodigious and tragic Andreas Floer … . the book is exceptionally well written. Indeed, this is a very good book on a beautiful and important subject and will richly repay those who take the time to work through it.” (Michael Berg, MAA Reviews, February, 2014)Table of ContentsIntroduction to Part I.- Morse Functions.- Pseudo-Gradients.- The Morse Complex.- Morse Homology, Applications.- Introduction to Part II.- What You Need To Know About Symplectic Geometry.- The Arnold Conjecture and the Floer Equation.- The Maslov Index.- Linearization and Transversality.- Spaces of Trajectories.- From Floer To Morse.- Floer Homology: Invariance.- Elliptic Regularity.- Technical Lemmas.- Exercises for the Second Part.- Appendices: What You Need to Know to Read This Book.
£71.99
Springer First Steps in Differential Geometry
Book SynopsisBasic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.Trade ReviewFrom the book reviews:“This books presents an alternative route, aiming to provide the student with an introduction not only to Riemannian geometry, but also to contact and symplectic geometry. … the book is leavened with an excellent collection of illustrative examples, and a wealth of exercises on which students can hone their skills. Each chapter also includes a short guide to further reading on the topic with a helpful brief commentary on the suggestions.” (Robert J. Low, Mathematical Reviews, May, 2014)“This book is a distinctive and ambitious effort to bring modern notions of differential geometry to undergraduates. … Mclnerney’s writing is well constructed and very clear … . Summing Up: Recommended. Upper-division undergraduates and graduate students.” (S. J. Colley, Choice, Vol. 51 (8), April, 2014)“The author does make a considerable effort to keep things as accessible as possible, with fairly detailed explanations, extensive motivational discussions and homework problems … . this book provides a different way of looking at the subject of differential geometry, one that is more modern and sophisticated than is provided by many of the standard undergraduate texts and which will certainly do a good job of preparing the student for additional work in this area down the road.” (Mark Hunacek, MAA Reviews, January, 2014)“This text provides an early and broad view of geometry to mathematical students … . Altogether, this book is easy to read because there are plenty of figures, examples and exercises which make it intuitive and perfect for undergraduate students.” (Teresa Arias-Marco, zbMATH, Vol. 1283, 2014)Table of ContentsBasic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.
£51.29
Springer Us Submanifold Theory Beyond an Introduction
Book SynopsisSeveral relevant classes of submanifolds are also discussed, including constant curvature submanifolds, submanifolds of nonpositive extrinsic curvature, conformally flat submanifolds and real Kaehler submanifolds.Trade Review“I can recommend this book to anyone interested in submanifold theory: from students with a background in basic Riemannian geometry to experienced researchers in the field.” (Joeri Van der Veken, Mathematical Reviews, January, 2021)Table of ContentsThe basic equations of a submanifold.- Reduction of codimension.- Minimal submanifolds.- Local rigidity of submanifolds.- Constant curvature submanifolds.- Submanifolds with nonpositive extrinsic curvature.- Submanifolds with relative nullity.- Isometric immersions of Riemannian products.- Conformal immersions.- Isometric immersions of warped products.- The Sbrana-Cartan hypersurfaces.- Genuine deformations.- Deformations of complete submanifolds.- Innitesimal bendings.- Real Kaehler submanifolds.- Conformally at submanifolds.- Conformally deformable hypersurfaces.- Vector bundles.
£62.99
