Differential and Riemannian geometry Books

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.

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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.

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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

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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.

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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 *

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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

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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

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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

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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.

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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.

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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

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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.

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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.

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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

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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.

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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.

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Book SynopsisThe latest volume in the New York Times–bestselling physics series explains Einstein’s masterpiece: the general theory of relativity He taught us classical mechanics, quantum mechanics, and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein’s general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations, and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler). An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe’s real structure.  

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Book SynopsisThis collection presents the major mathematical works of Isadore Singer, selected by Singer himself, and organized thematically into three volumes: 1. Functional analysis, differential geometry and eigenvalues 2. Index theory 3. Gauge theory and physics Each volume begins with a commentary (and in the first volume, a short biography of Singer), and then presents the works on its theme in roughly chronological order.

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Book SynopsisDifferential geometry is a mathematical discipline that uses the methods of differential and integral calculus to study problems in geometry. Graph theory is also a growing area in mathematical research. In mathematics and computer science, graph theory is the study of mathematical structures used to model pairwise relations between objects from a certain collection. This book presents various theories and applications in both of these mathematical fields. Included are the concepts of dominating sets, one of the most widely studied concepts in graph theory, some current developments of graph theory in the fields of planar linkage mechanisms and geared linkage mechanisms, lie algebras and the application of CR Hamiltonian flows to the deformation theory of CR structures.

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Book SynopsisThis book offers a first taste of the theory of Lie groups, focusing mainly on matrix groups: closed subgroups of real and complex general linear groups. The first part studies examples and describes classical families of simply connected compact groups. The second section introduces the idea of a lie group and explores the associated notion of a homogeneous space using orbits of smooth actions. The emphasis throughout is on accessibility. Trade ReviewFrom the reviews of the first edition: MATHEMATICAL REVIEWS "This excellent book gives an easy introduction to the theory of Lie groups and Lie algebras by restricting the material to real and complex matrix groups. This provides the reader not only with a wealth of examples, but it also makes the key concepts much more concrete. This combination makes the material in this book more easily accessible for the readers with a limited background…The book is very easy to read and suitable for an elementary course in Lie theory aimed at advanced undergraduates or beginning graduate students…To summarize, this is a well-written book, which is highly suited as an introductory text for beginning graduate students without much background in differential geometry or for advanced undergraduates. It is a welcome addition to the literature in Lie theory." "This book is an introduction to Lie group theory with focus on the matrix case. … This book can be recommended to students, making Lie group theory more accessible to them." (A. Akutowicz, Zentralblatt MATH, Vol. 1009, 2003)Table of ContentsI. Basic Ideas and Examples.- 1. Real and Complex Matrix Groups.- 2. Exponentials, Differential Equations and One-parameter Subgroups.- 3. Tangent Spaces and Lie Algebras.- 4. Algebras, Quaternions and Quaternionic Symplectic Groups.- 5. Clifford Algebras and Spinor Groups.- 6. Lorentz Groups.- II. Matrix Groups as Lie Groups.- 7. Lie Groups.- 8. Homogeneous Spaces.- 9. Connectivity of Matrix Groups.- III. Compact Connected Lie Groups and their Classification.- 10. Maximal Tori in Compact Connected Lie Groups.- 11. Semi-simple Factorisation.- 12. Roots Systems, Weyl Groups and Dynkin Diagrams.- Hints and Solutions to Selected Exercises.

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Book SynopsisThis book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.Trade Review“The present book is well written. It is very useful to researchers in differential geometry who are interested in non-commutative geometry. It provides motivations for tudying non commutative geometry.” (Ion Mihai, zbMATH 1458.58001, 2021)Table of Contents1. Part I Spaces, bundles and characteristic classes in differential geometry.- 2. Part II Non-commutative differential geometry.- 3. Part III Index Theorems.- 4. Part IV Prospects in Index Theory. Part V.- 5. Non-commutative topology.

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Book SynopsisThis textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.Trade Review“The student will benefit from the many illustrative examples worked out in the book. … The author succeeds, and the hope is that after working on some of the examples presented, the student will want to explore further applications. Additionally, the instructor may also find inspiration for individual study topics that don’t require extensive prerequisites.” (Valentin Keyantuo, Mathematical Reviews, February, 2023)“This is an excellent textbook, which shall be a very useful tool for anyone who is oriented to the applications of functional analysis, especially to partial differential equations.” (Panagiotis Koumantos, zbMATH 1444.47001, 2020)Table of Contents1. Introduction.- 2. Hilbert Spaces.- 3. Operators.- 4. Spectrum and Resolvent.- 5. The Spectral Theorem.- 6. The Laplacian with Boundary Conditions.- 7. Schrödinger Operators.- 8. Operators on Graphs.- 9. Spectral Theory on Manifolds.- A. Background Material.- References.- Index.

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Book SynopsisThis textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.Trade Review“The book … is intended ‘for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly engineering students, especially in computer science.’ … The text’s coverage is extensive, its exposition clear throughout, and the color illustrations helpful. The authors are also familiar with many texts at a comparable level and have drawn on them in several places to include some of the most insightful proofs already in the literature.” (Jer-Chin Chuang, MAA Reviews, October 4, 2021)“The book is intended for incremental study and covers both basic concepts and more advanced ones. The former are thoroughly supported with theory and examples, and the latter are backed up with extensive reading lists and references. … Thanks to its design and approach style this is a timely and much needed addition that enables interdisciplinary bridges and the discovery of new applications for differential geometry.” (Corina Mohorian, zbMATH 1453.53001, 2021)Table of Contents1. The Matrix Exponential; Some Matrix Lie Groups.- 2. Adjoint Representations and the Derivative of exp.- 3. Introduction to Manifolds and Lie Groups.- 4. Groups and Group Actions.- 5. The Lorentz Groups ⊛.- 6. The Structure of O(p,q) and SO(p, q).- 7. Manifolds, Tangent Spaces, Cotangent Spaces.- 8. Construction of Manifolds From Gluing Data ⊛.- 9. Vector Fields, Integral Curves, Flows.- 10. Partitions of Unity, Covering Maps ⊛.- 11. Basic Analysis: Review of Series and Derivatives.- 12. A Review of Point Set Topology.-13. Riemannian Metrics, Riemannian Manifolds.- 14. Connections on Manifolds.- 15. Geodesics on Riemannian Manifolds.- 16. Curvature in Riemannian Manifolds.- 17. Isometries, Submersions, Killing Vector Fields.- 18. Lie Groups, Lie Algebra, Exponential Map.- 19. The Derivative of exp and Dynkin's Formula ⊛.- 20. Metrics, Connections, and Curvature of Lie Groups.- 21. The Log-Euclidean Framework.- 22. Manifolds Arising from Group Actions.

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Book SynopsisThis book contains all research papers published by the distinguished Brazilian mathematician Elon Lima. It includes the papers from his PhD thesis on homotopy theory, which are hard to find elsewhere. Elon Lima wrote more than 40 books in the field of topology and dynamical systems. He was a profound mathematician with a genuine vocation to teach and write mathematics.Table of ContentsComments on some mathematical contributions of Elon Lima.- The Spanier-Whitehead duality in new homotopy categories.- Stable Postnikov invariants and their duals.- Commuting vector fields on 2-manifolds.- On the local triviality of the restriction map for embeddings.- Commuting vector fields on S2.- Common singularities of commuting vector fields on 2-manifolds.- Commuting vector fields on S3.- Isometric immersions with semi-definite second quadratic forms.- Immersions of manifolds with non-negative sectional curvatures.- Orientability of smooth hypersurfaces and the Jordan-Brouwer separation theorem.- The Jordan-Brouwer separation theorem for smooth hypersurfaces.

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Book SynopsisThis book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics.The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.Trade Review“The reader is assumed to have some background in topology and differential geometry. The book is a continuation of the authors’ work in extrinsic geometry and thus provides a useful reference for researchers in this field.” (Emanuel-Ciprian Cismaş, zbMATH 1479.53002, 2022)Table of ContentsPreface.- 1. Preliminaries.- 2. Integral formulas.- 3. Prescribing the mean curvature.- 4. Variational formulae.- 5. Extrinsic Geometric flows.- References.- Index.

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Book SynopsisThis volume originated in talks given in Cortona at the conference "Geometric aspects of harmonic analysis" held in honor of the 70th birthday of Fulvio Ricci. It presents timely syntheses of several major fields of mathematics as well as original research articles contributed by some of the finest mathematicians working in these areas. The subjects dealt with are topics of current interest in closely interrelated areas of Fourier analysis, singular integral operators, oscillatory integral operators, partial differential equations, multilinear harmonic analysis, and several complex variables.The work is addressed to researchers in the field.Table of Contents- An Extension Problem and Hardy Type Inequalities for the Grushin Operator. - Sharp Local Smoothing Estimates for Fourier Integral Operators. - On the Hardy–Littlewood Maximal Functions in High Dimensions: Continuous and Discrete Perspective. - Potential Spaces on Lie Groups. - On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid. - On Young’s Convolution Inequality for Heisenberg Groups. - Young’s Inequality Sharpened. - Strongly Singular Integrals on Stratified Groups. - Singular Brascamp–Lieb: A Survey. - On the Restriction of Laplace–Beltrami Eigenfunctions and Cantor-Type Sets. - Basis Properties of the Haar System in Limiting Besov Spaces. - Obstacle Problems Generated by the Estimates of Square Function. - Of Commutators and Jacobians. - On Regularity and Irregularity of Certain Holomorphic Singular Integral Operators.

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Book SynopsisThis monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area. Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner–Weitzenböck formula and the corresponding Bochner inequality, gradient estimates, Bakry–Ledoux’s Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement. Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.Trade Review“Finsler geometry is an active area of research in mathematics and has led to numerous real-world applications. This book is a comprehensive introduction to Finsler geometry and its applications. It covers the basic concepts of this geometry. More intuitively, this book provides an accessible introduction to recent developments in comparison geometry and geometric analysis on Finsler manifolds. … this book offers a valuable perspective for those familiar with comparison geometry and geometric analysis.” (Behroz Bidabad, Mathematical Reviews, May, 2023)Table of ContentsI Foundations of Finsler Geometry.- 1. Warm-up: Norms and inner products.- 2. Finsler manifolds.- 3. Properties of geodesics.- 4. Covariant derivatives.- 5. Curvature.- 6. Examples of Finsler manifolds.- 7. Variation formulas for arclength.- 8. Some comparison theorems.- II Geometry and analysis of weighted Ricci curvature.- 9. Weighted Ricci curvature.- 10. Examples of measured Finsler manifolds.- 11. The nonlinear Laplacian.- 12. The Bochner-Weitzenbock formula.- 13. Nonlinear heat flow.- 14. Gradient estimates.- 15. Bakry-Ledoux isoperimetric inequality.- 16. Functional inequalities.- III Further topics.- 17. Splitting theorems.- 18. Curvature-dimension condition.- 19. Needle decompositions.

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Book SynopsisThis book consists of contributions from the participants of the Abel Symposium 2019 held in Ålesund, Norway. It was centered about applications of the ideas of symmetry and invariance, including equivalence and deformation theory of geometric structures, classification of differential invariants and invariant differential operators, integrability analysis of equations of mathematical physics, progress in parabolic geometry and mathematical aspects of general relativity.The chapters are written by leading international researchers, and consist of both survey and research articles. The book gives the reader an insight into the current research in differential geometry and Lie theory, as well as applications of these topics, in particular to general relativity and string theory.Table of ContentsFour-dimensional homogeneous generalizations of Einstein Metrics.- Conformal and isometric embeddings of gravitational instantons.- Recent results on closed G2-structures, by Anna Fino and Alberto Raffero.- Almost Zoll affine surfaces.- Distinguished curves and fist integrals on Poincare-Einstein and other conformally singular geometries.- A car as parabolic geometry.- Legendrian cone structures and contact prolongations.- The search for solitons on homogeneous spaces.- On Ricci negative Lie groups.- Semi-Riemannian cones.- Building new Einstein spaces by deforming symmetric Einstein spaces.- Remarks on highly supersymmetric backgrounds of 11-dimensional supergravity.- Krichever-Novikov type algebras.

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Book SynopsisThis book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces.The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.Trade Review“All chapters are supplemented with solutions of the problems scattered throughout the text. Designed as a text for a lecturer course on the subject, it is perfect and can be recommended for students interested in this classical field.” (Ivailo. M. Mladenov, zbMATH 1498.53001, 2022)Table of ContentsCurves in the Plane.- Curves in Space.- Surfaces in Space.- Hypersurfaces in Rn+1.- Connections.- Riemannian Manifolds.- Lie Groups.- Comparison Theorems.- Curvature and Topology.- Laplacian.- Appendix.- Bibliography.- Index.

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Book SynopsisThis book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces.The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.Trade Review“All chapters are supplemented with solutions of the problems scattered throughout the text. Designed as a text for a lecturer course on the subject, it is perfect and can be recommended for students interested in this classical field.” (Ivailo. M. Mladenov, zbMATH 1498.53001, 2022)Table of ContentsCurves in the Plane.- Curves in Space.- Surfaces in Space.- Hypersurfaces in Rn+1.- Connections.- Riemannian Manifolds.- Lie Groups.- Comparison Theorems.- Curvature and Topology.- Laplacian.- Appendix.- Bibliography.- Index.

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