Differential and Riemannian geometry Books
Basic Books General Relativity: The Theoretical Minimum
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.
£24.00
International Press of Boston Inc Selected Works of Isadore Singer: Volume 2: Index
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.
£42.95
Nova Science Publishers Inc Emerging Topics on Differential Geometry & Graph
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.
£116.24
Society for Industrial & Applied Mathematics,U.S. Riemann Problems and Jupyter Solutions
Book SynopsisThis book addresses an important class of mathematical problems (the Riemann problem) for first-order hyperbolic partial differential equations (PDEs), which arise when modeling wave propagation in applications such as fluid dynamics, traffic flow, acoustics, and elasticity.It covers the fundamental ideas related to classical Riemann solutions, including their special structure and the types of waves that arise, as well as the ideas behind fast approximate solvers for the Riemann problem.The emphasis is on the general ideas, but each chapter delves into a particular application. The book is available in electronic form as a collection of Jupyter notebooks that contain executable computer code and interactive figures and animations.
£51.85
Nova Science Publishers Inc Quantum Mechanics in Spaces of Constant Curvature
Book Synopsis
£152.99
Lulu.com Basic procedures in ordinary differential
Book Synopsis
£14.07
World Scientific Europe Ltd Guide To Lie Systems With Compatible Geometric
Book SynopsisThe book presents a comprehensive guide to the study of Lie systems from the fundamentals of differential geometry to the development of contemporary research topics. It embraces several basic topics on differential geometry and the study of geometric structures while developing known applications in the theory of Lie systems. The book also includes a brief exploration of the applications of Lie systems to superequations, discrete systems, and partial differential equations.Offering a complete overview from the topic's foundations to the present, this book is an ideal resource for Physics and Mathematics students, doctoral students and researchers.
£121.50
Imperial College Press Geometric Realizations Of Curvature
Book SynopsisA central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including the latest developments and proofs. Even though many authors have worked in this area in recent years, many fundamental questions still remain unanswered. Many studies begin by first working purely algebraically and then later progressing onto the geometric setting and it has been found that many questions in differential geometry can be phrased as problems involving the geometric realization of curvature. Curvature decompositions are central to all investigations in this area. The authors present numerous results including the Singer-Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri-Vanhecke decomposition, the Gray-Hervella decomposition and the De Smedt decomposition. They then proceed to draw appropriate geometric conclusions from these decompositions.The book organizes, in one coherent volume, the results of research completed by many different investigators over the past 30 years. Complete proofs are given of results that are often only outlined in the original publications. Whereas the original results are usually in the positive definite (Riemannian setting), here the authors extend the results to the pseudo-Riemannian setting and then further, in a complex framework, to para-Hermitian geometry as well. In addition to that, new results are obtained as well, making this an ideal text for anyone wishing to further their knowledge of the science of curvature.Table of ContentsIntroduction and Statement of Results; Representation Theory; Connections, Curvature, and Differential Geometry; Affine Geometry; Kahler and Para-Kahler Geometry Affine Geometry; Riemannian Geometry; Complex and Para-Complex Geometry; Hyper Complex Geometry and Other Questions.
£82.65
Springer London Ltd Matrix Groups: An Introduction to Lie Group
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.
£34.94
Imperial College Press Introduction To Differential Manifolds, An
Book SynopsisThis invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.Trade Review"This book is an excellent introductory text into the theory of differential manifolds with a carefully thought out and tested structure and a sufficient supply of exercises and their solutions. It does not only guide the reader gently into the depths of the theory of differential manifolds but also careful on giving advice how one can place the information in the right context. It is certainly written in the best traditions of great Cambridge mathematics." Acta Scientiarum MathematicarumTable of ContentsDifferential Manifolds and Differentiable Maps; The Derivatives of Differentiable Maps; Fibre Bundles; Differential Forms and Integration; The Exterior Derivative; de Rham Cohomology; Degrees, Indices and Related Topics; Lie Groups; A Rapid Course in Differential Analysis; Solutions to the Exercises; Guide to the Literature.
£57.95
Imperial College Press Introduction To Differential Manifolds, An
Book SynopsisThis invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.Trade Review"This book is an excellent introductory text into the theory of differential manifolds with a carefully thought out and tested structure and a sufficient supply of exercises and their solutions. It does not only guide the reader gently into the depths of the theory of differential manifolds but also careful on giving advice how one can place the information in the right context. It is certainly written in the best traditions of great Cambridge mathematics." Acta Scientiarum MathematicarumTable of ContentsDifferential Manifolds and Differentiable Maps; The Derivatives of Differentiable Maps; Fibre Bundles; Differential Forms and Integration; The Exterior Derivative; de Rham Cohomology; Degrees, Indices and Related Topics; Lie Groups; A Rapid Course in Differential Analysis; Solutions to the Exercises; Guide to the Literature.
£35.15
Imperial College Press Geometry Of Curvature Homogeneous
Book SynopsisPseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and Stanilov-Tsankov-Videv theory.Table of ContentsThe Geometry of the Riemann Curvature Tensor; Curvature Homogeneous Generalized Plane Wave Manifolds; Other Pseudo-Riemannian Manifolds; The Curvature Tensor; Complex Osserman Algebraic Curvature Tensors; Stanilov-Tsankov Theory.
£126.90
Minkowski Institute Press Differential Geometry: 1972 Lecture Notes
£13.66
Springer Nature Switzerland AG From Differential Geometry to Non-commutative
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.
£104.49
Springer Nature Switzerland AG Spectral Theory: Basic Concepts and Applications
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.
£49.49
Springer Nature Switzerland AG Differential Geometry and Lie Groups: A
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.
£58.49
Springer Nature Switzerland AG Elon Lima - Selected Papers
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.
£40.49
Springer Nature Switzerland AG Extrinsic Geometry of Foliations
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.
£66.49
Springer Nature Switzerland AG Geometric Aspects of Harmonic Analysis
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.
£151.99
Springer Nature Switzerland AG Comparison Finsler Geometry
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.
£85.49
Springer Nature Switzerland AG Geometry, Lie Theory and Applications: The Abel
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.
£134.99
Springer Nature Switzerland AG Differential Geometry
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.
£48.74
Springer Nature Switzerland AG Differential Geometry
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.
£37.49
Springer Nature Switzerland AG Periodic Monopoles and Difference Modules
Book SynopsisThis book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized Cherkis–Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of Kobayashi–Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity.The theory of periodic monopoles of GCK type has applications to Yang–Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given, including major generalizations of results due to Charbonneau, Cherkis, Hurtubise, Kapustin, and others, and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette, Donaldson, Hitchin and Simpson.This work will be of interest to graduate students and researchers in differential and algebraic geometry, as well as in mathematical physics.Table of Contents. - Introduction. - Preliminaries. - Formal Difference Modules and Good Parabolic Structure. - Filtered Bundles. - Basic Examples of Monopoles Around Infinity. - Asymptotic Behaviour of Periodic Monopoles Around Infinity. - The Filtered Bundles Associated with Periodic Monopoles. - Global Periodic Monopoles of Rank One. - Global Periodic Monopoles and Filtered Difference Modules. - Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix).
£49.49
Springer International Publishing AG Introduction to Geometric Control
Book SynopsisThis text is an enhanced, English version of the Russian edition, published in early 2021 and is appropriate for an introductory course in geometric control theory. The concise presentation provides an accessible treatment of the subject for advanced undergraduate and graduate students in theoretical and applied mathematics, as well as to experts in classic control theory for whom geometric methods may be introduced. Theory is accompanied by characteristic examples such as stopping a train, motion of mobile robot, Euler elasticae, Dido's problem, and rolling of the sphere on the plane. Quick foundations to some recent topics of interest like control on Lie groups and sub-Riemannian geometry are included. Prerequisites include only a basic knowledge of calculus, linear algebra, and ODEs; preliminary knowledge of control theory is not assumed. The applications problems-oriented approach discusses core subjects and encourages the reader to solve related challenges independently. Highly-motivated readers can acquire working knowledge of geometric control techniques and progress to studying control problems and more comprehensive books on their own. Selected sections provide exercises to assist in deeper understanding of the material.Controllability and optimal control problems are considered for nonlinear nonholonomic systems on smooth manifolds, in particular, on Lie groups. For the controllability problem, the following questions are considered: controllability of linear systems, local controllability of nonlinear systems, Nagano–Sussmann Orbit theorem, Rashevskii–Chow theorem, Krener's theorem. For the optimal control problem, Filippov's theorem is stated, invariant formulation of Pontryagin maximum principle on manifolds is given, second-order optimality conditions are discussed, and the sub-Riemannian problem is studied in detail. Pontryagin maximum principle is proved for sub-Riemannian problems, solution to the sub-Riemannian problems on the Heisenberg group, the group of motions of the plane, and the Engel group is described.Table of Contents1. Introduction.- 2. Controllability problem.- 3. Optimal control problem.- 4. Solution to optimal control problems.- 5. Conclusion.- A. Elliptic integrals, functions and equation of pendulum.- Bibliography and further reading.- Index.
£52.24
Springer International Publishing AG Linear Fractional Transformations: An Illustrated
Book SynopsisThe principle aim of this unique text is to illuminate the beauty of the subject both with abstractions like proofs and mathematical text, and with visuals, such as abundant illustrations and diagrams. With few mathematical prerequisites, geometry is presented through the lens of linear fractional transformations. The exposition is motivational and the well-placed examples and exercises give students ample opportunity to pause and digest the material. The subject builds from the fundamentals of Euclidean geometry, to inversive geometry, and, finally, to hyperbolic geometry at the end. Throughout, the author aims to express the underlying philosophy behind the definitions and mathematical reasoning. This text may be used as primary for an undergraduate geometry course or a freshman seminar in geometry, or as supplemental to instructors in their undergraduate courses in complex analysis, algebra, and number theory. There are elective courses that bring together seemingly disparate topics and this text would be a welcome accompaniment.Table of ContentsMotivation.- I Euclidean and Inversive Geometry.- Euclidean Isometries and Similarities.- Inversive Geometry.- Applications of Inversive Geometry.- II Non-Euclidean Geometry.- Spherical Geometry.- Appendix: Set Theory.
£40.49
Springer International Publishing AG Manifolds, Vector Fields, and Differential Forms:
Book SynopsisThis textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum. Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging. The book contains more material than can be covered in a single semester, and detailed suggestions for instructors are provided in the Preface.Trade Review“This book is intended to be a modern introduction to the basics of differential geometry, accessible to undergraduate and master students. From my point of view, this goal is achieved, the book being very well structured and supported by illustrative examples and problems. … this book will be of great interest for undergraduate students, master students, and also helpful for instructors.” (Gabriel Eduard Vilc, zbMATH 1522.53001, 2023)Table of Contents1. Introduction2. Manifolds3. Smooth maps4. Submanifolds5. Tangent spaces6. Vector fields7. Differential forms8. Integration9. Vector bundlesNotions from set theoryNotions from algebraTopological properties of manifoldsHints and answers to in-text questionsReferencesList of SymbolsIndex
£35.99
Springer International Publishing AG The Volume of Vector Fields on Riemannian
Book SynopsisThis book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subject’s introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs.A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); a thorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three.Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.Table of Contents- 1. Introduction. - 2. Minimal Sections of Tensor Bundles. - 3. Minimal Vector Fields of Constant Length on the Odd-Dimensional Spheres. - 4. Vector Fields of Constant Length of Minimum Volume on the Odd-Dimensional Spherical Space Forms. - 5. Vector Fields of Constant Length on Punctured Spheres.
£44.99
Springer International Publishing AG Differential Geometry
Book Synopsis
£40.49
Birkhäuser Advances in Poisson Geometry
Book Synopsis- Part I: Introduction to Poisson Geometry.- 1. A brief Introduction to Poisson Geometry.- Part II: Wonderful Varieties.- 2. Wonderful Varieties with a View Towards Poisson Geometry.- Part III: An Invitation to Singular Foliations.- 3. What is a singular foliation?.- 4. Canonical geometric and algebraic structures hidden behind a singular foliation.- 5. State of the Art and open questions.
£44.99
Springer Geometric Deformations of Discriminants and
Book SynopsisChapter 1. Map-germs from the plane to the plane.- Chapter 2. Geometric deformations of discriminants.- Chapter 3. Geometric deformations of the fold and cusp.- Chapter 4. Ae-codimension 1 singularities.- Chapter 5. Ae-codimension 2 singularities.- Chapter 6. Apparent contours.- Chapter 7. Geometric invariants.
£59.39
Springer Exceptional Lie groups
Book SynopsisChapter 1. Exceptional Lie group G2.- Chapter 2. Exceptional Lie group F4.- Chapter 3. Exceptional Lie group E6.- Chapter 4. Exceptional Lie group E7.- Chapter 5. Exceptional Lie group E8.
£58.49
Springer Discrete Isothermic Surfaces in Lie Sphere
Book SynopsisChapter 1. Introduction.- Chapter 2.Isothermic surfaces in Möbius geometry.- Chapter 3. From smooth to discrete via permutability.- Chapter 4. Discrete Isothermic surfaces.- Chapter 5. ?-surfaces in Lie sphere geometry.- Chapter 6. Integrability of ?-surfaces via isothermicity.- Chapter 7. Discrete ?-surfaces.
£61.74
De Gruyter Riemannian Geometry
Book SynopsisThe series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob. Titles in planning include Flavia Smarazzo and Alberto Tesei, Measure Theory: Radon Measures, Young Measures, and Applications to Parabolic Problems (2019) Elena Cordero and Luigi Rodino, Time-Frequency Analysis of Operators (2019) Mark M. Meerschaert, Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational Models for Fractional Calculus, second edition (2020) Mariusz Lemańczyk, Ergodic Theory: Spectral Theory, Joinings, and Their Applications (2020) Marco Abate, Holomorphic Dynamics on Hyperbolic Complex Manifolds (2021) Miroslava Antić, Joeri Van der Veken, and Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds of Almost Complex Spaces and Almost Product Spaces (2021) Kai Liu, Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference Equations (2021) Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi Miyanishi, Affine Space Fibrations (2022)
£138.22
De Gruyter Global Affine Differential Geometry of Hypersurfaces
Book SynopsisThis book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces. The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.
£123.98
de Gruyter Weighted Morrey Spaces
Book Synopsis
£116.85
Walter de Gruyter Differential Geometry
Book Synopsis
£68.00
Springer International Publishing AG Elementary Symplectic Topology and Mechanics
Book SynopsisThis is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.Trade Review“This book brings the reader from the basics of symplectic topology through several concepts and issues relative to Hamiltonian and Lagrangian systems, Hamilton-Jacobi equations, the calculus of variations, and Morse and Lusternik-Schnirelman theories. … The book seems best suited to researchers and students from graduate level onwards.” (Maria Letizia Bertotti, Mathematical Reviews, August, 2015)Table of ContentsBeginning.- Notes on Differential Geometry.- Symplectic Manifolds.- Poisson brackets environment.- Cauchy Problem for H-J equations.- Calculus of Variations and Conjugate Points.- Asymptotic Theory of Oscillating Integrals.- Lusternik-Schnirelman and Morse.- Finite Exact Reductions.- Other instances.- Bibliography.
£47.49
Birkhauser Verlag AG Space-Time Algebra
Book SynopsisThis small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future.At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas. These same techniques, in the form of the ‘Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, and enables physicists to understand topics in engineering, and engineers to understand topics in physics (including aspects in frontier areas), in a way which no other single mathematical system could hope to make possible.There is another aspect to Geometric Algebra, which is less tangible, and goes beyond questions of mathematical power and range. This is the remarkable insight it gives to physical problems, and the way it constantly suggests new features of the physics itself, not just the mathematics. Examples of this are peppered throughout ‘Space-Time Algebra’, despite its short length, and some of them are effectively still research topics for the future.From the Foreward by Anthony LasenbyTable of ContentsPreface to the Second Edition.- Introduction.- Part I:Geometric Algebra.- 1.Intrepretation of Clifford Algebra.- 2.Definition of Clifford Algebra.- 3.Inner and Outer Products.- 4.Structure of Clifford Algebra.- 5.Reversion, Scalar Product.- 6.The Algebra of Space.- 7.The Algebra of Space-Time.- Part II:Electrodynamics.- 8.Maxwell's Equation.- 9.Stress-Energy Vectors.- 10.Invariants .- 11. Free Fields.- Part III:Dirac Fields.- 12.Spinors.- 13.Dirac's Equation.- 14.Conserved Currents.- 15.C, P, T.- Part IV:Lorentz Transformations.- 16.Reflections and Rotations.- 17.Coordinate Transformations.- 18.Timelike Rotations.- 19.Scalar Product.- Part V:Geometric Calculus.- 20.Differentiation.- 21.Coordinate Transformations.- 22.Integration.- 23.Global and Local Relativity.- 24.Gauge Transformation and Spinor Derivatives.- Conclusion.- Appendices.- A.Bases and Pseudoscalars.- B.Some Theorems.- C.Composition of Spacial Rotations.- D.Matrix Representation of the Pauli Algebra.
£94.99
Springer International Publishing AG Differential Geometry of Curves and Surfaces
Book SynopsisThis is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging.Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. Trade Review“This is the first textbook on mathematics that I see printed in color. … This book is not a usual textbook, but a very well written introduction to differential geometry, and the colors really help the reader in understanding the figures and navigating through the text. … this book will surely serve very well for students who want to learn differential geometry from the ground up no matter what their main learning goal is.” (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 84 (1-2), 2018)“This book is perfect for undergraduate students. ... There is also plenty of figures, examples, exercises and applications which make the differential geometry of curves and surfaces so interesting and intuitive. The author uses a rich variety of colours and techniques that help to clarify difficult abstract concepts.” (Teresa Arias-Marco, zbMATH 1375.53001, 2018)“This is a visually appealing book, replete with many diagrams, lots of them in full color. … the author’s writing style is extremely clear and well-motivated. … this is still the book I would use as a text for a beginning course on this subject. It would not surprise me if it quickly becomes the market leader.” (Mark Hunacek, MAA Reviews, July, 2017) Table of ContentsIntroduction.- Curves.- Additional topics in curves.- Surfaces.- The curvature of a surface.- Geodesics.- The Gauss–Bonnet theorem.- Appendix A: The topology of subsets of Rn.- Recommended excursions.- Index.
£49.49
Springer International Publishing AG Differential Geometry: Connections, Curvature,
Book SynopsisThis text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text.Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included.Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.Trade Review“The textbook is a concise and well organized treatment of characteristic classes on principal bundles. It is characterized by a right balance between rigor and simplicity. It should be in every mathematician's arsenal and take its place in any mathematical library.” (Nabil L. Youssef, zbMATH 1383.53001, 2018)Table of ContentsPreface.- Chapter 1. Curvature and Vector Fields.- 1. Riemannian Manifolds.- 2. Curves.- 3. Surfaces in Space.- 4. Directional Derivative in Euclidean Space.- 5. The Shape Operator.- 6. Affine Connections.- 7. Vector Bundles.- 8. Gauss's Theorema Egregium.- 9. Generalizations to Hypersurfaces in Rn+1.- Chapter 2. Curvature and Differential Forms.- 10. Connections on a Vector Bundle.- 11. Connection, Curvature, and Torsion Forms.- 12. The Theorema Egregium Using Forms.- Chapter 3. Geodesics.- 13. More on Affine Connections.- 14. Geodesics.- 15. Exponential Maps.- 16. Distance and Volume.- 17. The Gauss-Bonnet Theorem.- Chapter 4. Tools from Algebra and Topology.- 18. The Tensor Product and the Dual Module.- 19. The Exterior Power.- 20. Operations on Vector Bundles.- 21. Vector-Valued Forms.- Chapter 5. Vector Bundles and Characteristic Classes.- 22. Connections and Curvature Again.- 23. Characteristic Classes.- 24. Pontrjagin Classes.- 25. The Euler Class and Chern Classes.- 26. Some Applications of Characteristic Classes.- Chapter 6. Principal Bundles and Characteristic Classes.- 27. Principal Bundles.- 28. Connections on a Principal Bundle.- 29. Horizontal Distributions on a Frame Bundle.- 30. Curvature on a Principal Bundle.- 31. Covariant Derivative on a Principal Bundle.- 32. Character Classes of Principal Bundles.- A. Manifolds.- B. Invariant Polynomials.- Hints and Solutions to Selected End-of-Section Problems.- List of Notations.- References.- Index.
£49.49
Springer International Publishing AG Selected Works I
Book SynopsisThis is a two-volume collection presenting the selected works of Herbert Busemann, one of the leading geometers of the twentieth century and one of the main founders of metric geometry, convexity theory and convexity in metric spaces. Busemann also did substantial work (probably the most important) on Hilbert’s Problem IV. These collected works include Busemann’s most important published articles on these topics. Volume I of the collection features Busemann’s papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces. Volume II includes Busemann’s papers on convexity and integral geometry, on Hilbert’s Problem IV, and other papers on miscellaneous subjects. Each volume offers biographical documents and introductory essays on Busemann’s work, documents from his correspondence and introductory essays written by leading specialists on Busemann’s work. They are a valuable resource for researchers in synthetic and metric geometry, convexity theory and the foundations of geometry. Table of ContentsPreface.- Introduction to Volume I.- List of publications of Herbert Busemann.- Acknowledgements.- Essays.- A. Papadpoulos: Herbert Busemann (1905-1994).- A. Papadopoulos and M. Troyanov: On three early papers by Herbert Busemann on the foundations of geometry.- M. Troyanov: On Pasch's Axiom and Desargues' Theorem in Busemann's work.- V. N. Berestovskiy: Busemann's results, ideas, questions on locally compact homogeneous geodesic spaces.- A. Papadopoulos and S. Yamada: Busemann's problems on G-spaces.- Busemann's metric theory of timelike spaces.- A. Papadopoulos: Chronogeometry.- W. M. Boothby: Review of Busemann's book The geometry of Geodesics.- F. A. Ficken: Review of Busemann's book Metric Methods in Finsler Spaces and in the Foundations of Geometry.- Busemann's papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces.
£114.75
Springer International Publishing AG Introduction to Riemannian Manifolds
Book SynopsisThis text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.Trade Review“One interesting aspect of the book is the decision of which audience to target it towards. … Overall, this would make a very appropriate text for a graduate course, or a programme of individual study in Riemannian geometry, whether to give a thorough treatment of the fundamentals, or to introduce the more advanced topics in global geometry.” (Robert J. Low, Mathematical Reviews, November, 2019)“This material is carefully developed and several useful examples and exercises are included in each chapter. The reviewer’s belief is that this excellent edition will become soon a standard text for several graduate courses as well as an frequent citation in articles.” (Mircea Crâşmăreanu, zbMATH 1409.53001, 2019)Table of ContentsPreface.- 1. What Is Curvature?.- 2. Riemannian Metrics.- 3. Model Riemannian Manifolds.- 4. Connections.- 5. The Levi-Cevita Connection.- 6. Geodesics and Distance.- 7. Curvature.- 8. Riemannian Submanifolds.- 9. The Gauss–Bonnet Theorem.- 10. Jacobi Fields.- 11. Comparison Theory.- 12. Curvature and Topology.- Appendix A: Review of Smooth Manifolds.- Appendix B: Review of Tensors.- Appendix C: Review of Lie Groups.- References.- Notation Index.- Subject Index.
£49.49
Springer International Publishing AG A Brief Introduction to Berezin–Toeplitz
Book SynopsisThis text provides a comprehensive introduction to Berezin–Toeplitz operators on compact Kähler manifolds. The heart of the book is devoted to a proof of the main properties of these operators which have been playing a significant role in various areas of mathematics such as complex geometry, topological quantum field theory, integrable systems, and the study of links between symplectic topology and quantum mechanics. The book is carefully designed to supply graduate students with a unique accessibility to the subject. The first part contains a review of relevant material from complex geometry. Examples are presented with explicit detail and computation; prerequisites have been kept to a minimum. Readers are encouraged to enhance their understanding of the material by working through the many straightforward exercises. Trade Review“The book … represents an essential prerequisite for anyone who wants to work in the field. The author have managed to make it readable by non-specialists.” (Béchir Dali, zbMATH 1452.32002, 2021)Table of ContentsPreface.- 1. Introduction.- 2. A short introduction to Kähler manifolds.- 3. Complex line bundles with connections.- 4. Quantization of compact Kähler manifolds.- 5. Berezin–Toeplitz operators.- 6. Schwartz kernels.- 7. Asymptotics of the projector Pi_k.- 8. Proof of product and commutator estimates.- 9. Coherent states and norm correspondence.- A. The circle bundle point of view.- Bibliography.
£49.49
Birkhauser Verlag AG A Visual Introduction to Differential Forms and
Book SynopsisThis book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.Trade Review “The reviewer recommends young mathematics and physics majors to open the book and to keep it on their bookshelves. Indeed, the reviewer even envies young students who can study differential forms with such a fascinating book.” (Hirokazu Nishimura, zbMath 1419.58001, 2019)Table of Contents
£53.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Elementare Differentialgeometrie mit Maple
Book SynopsisIn dem Buch wird die Kurven- und Flächentheorie im 3-dimensionalen euklidischen Raum behandelt und ein Maple-Programmpaket auf einer CD zum konkreten Arbeiten geliefert. Mit einer neuen Programmerweiterung und dem theoretischen Hintergrund unter www.mi.uni-koeln.de/mi/Forschung/Reckziegel/diffgeo_ext/index.html.Table of ContentsDer Raum der elementaren Differentialgeometrie - Maple-Arbeitsmethoden im IR - Ebenen-Kurventheorie - Räumliche Kurventheorie - Einführung in die Flächentheorie - Modellierung von Flächen und Riemannschen Gebieten mit Maple - Äußere Geometrie von Flächen - Innere Geometrie von Flächen - Eine kurze Einführung in Maple
£26.59
Springer Fachmedien Wiesbaden Differentialgeometrie von Kurven und Flächen
Book SynopsisEs gibt in der Differentialgeometrie von Kurven und FJachen zwei Betrachtungsweisen. Die eine, die man klassische Differentialgeometrie nennen konnte, entstand zusammen mit den Anfangen der Differential-und Integralrechnung. Grob gesagt studiert die klassische Differentialgeometrie lokale Eigenschaften von Kurven und FHichen. Dabei verstehen wir unter lokalen Eigenschaften solche, die nur vom Verhalten der Kurve oder Flache in der Umgebung eines Punktes abhiingen. Die Methoden, die sich als fUr das Studium solcher Eigenschaften geeignet erwiesen haben, sind die Methoden der Differentialrechnung. Aus diesem Grund sind die in der Differentialgeometrie untersuchten Kurven und Flachen durch Funktionen definiert, die von einer gewissen Differenzierbarkeitsklasse sind. Die andere Betrachtungsweise ist die sogenannte globale Differentialgeometrie. Hierbei untersucht man den EinfluB lokaler Eigenschaften auf das Verhalten der gesamten Kurve oder Flache. Der interessanteste und reprasentativste Teil der klassischen Differentialgeometrie ist wohl die Untersuchung von Flachen. Beim Studium von Flachen treten jedoch in nattirlicher Weise einige 10k ale Eigenschaften von Kurven auf. Deshalb benutzen wir dieses erste Kapi tel, urn kurz auf Kurven einzugehen.Table of Contents1 Kurven.- 1.1 Einleitung.- 1.2 Parametrisierte Kurven.- 1.3 Reguläre Kurven. Bogenlänge.- 1.4 Das Vektorprodukt in ?3.- 1.5 Die lokale Theorie von Kurven, die nach der Bogenlänge parametrisiert sind.- 1.6 Die lokale kanonische Form.- 1.7 Globale Eigenschaften ebener Kurven.- 2 Reguläre Flächen.- 2.1 Einleitung.- 2.2 Reguläre Flächen. Urbilder regulärer Werte.- 2.3 Parameterwechsel. Differenzierbare Funktionen auf Flächen.- 2.4 Die Tangentialebene. Das Differential einer Abbildung.- 2.5 Die erste Fundamentalform. Flächeninhalt.- 2.6 Orientierung von Flächen.- 2.7 Eine Charakterisierung kompakter orientierbarer Flächen.- 2.8 Eine geometrische Definition des Flächeninhalts.- 3 Die Geometrie der Gauß-Abbildung.- 3.1 Einleitung.- 3.2 Die Definition der Gauß-Abbildung und ihre fundamentalen Eigenschaften.- 3.3 Die Gauß-Abbildung in lokalen Koordinaten.- 3.4 Vektorfelder.- 3.5 Regelflächen und Minimalflächen.- 4 Die innere Geometrie von Flächen.- 4.1 Einleitung.- 4.2 Isometrie. Konforme Abbildungen.- 4.3 Der Satz von Gauß und die Verträglichkeitsbedingungen.- 4.4 Parallelverschiebung. Geodätische.- 4.5 Der Satz von Gauß-Bonnet und seine Anwendungen.- 4.6 Die Exponentialabbildung. Geodätische Polarkoordinaten.- 4.7 Weitere Eigenschaften von Geodätischen. Konvexe Umgebungen.- Anhang: Beweise der Fundamentalsätze der lokalen Kurven-und Flächentheorie.- Hinweise und Lösungen.- Kommentiertes Literaturverzeichnis.- Namen-und Sachwortverzeichnis.
£44.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Riemannian Geometry
Book SynopsisThis book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.Trade ReviewFrom the reviews of the third edition: "This new edition maintains the clear written style of the original, including many illustrations … examples and exercises (most with solutions)." (Joseph E. Borzellino, Mathematical Reviews, 2005) "This book based on graduate course on Riemannian geometry … covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results … are treated in detail. … contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics … have been added and worked out in the same spirit." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004) "This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris. … Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples." (EMS Newsletter, December 2005) "The guiding line of this by now classic introduction to Riemannian geometry is an in-depth study of each newly introduced concept on the basis of a number of reoccurring well-chosen examples … . The book continues to be an excellent choice for an introduction to the central ideas of Riemannian geometry." (M. Kunzinger, Monatshefte für Mathematik, Vol. 147 (1), 2006)Table of Contents1 Differential manifolds.- 1.A From submanifolds to abstract manifolds.- 1.A.1 Submanifolds of Euclidean spaces.- 1.A.2 Abstract manifolds.- 1.A.3 Smooth maps.- 1.B The tangent bundle.- 1.B.1 Tangent space to a submanifold of Rn+k.- 1.B.2 The manifold of tangent vectors.- 1.B.3 Vector bundles.- 1.B.4 Tangent map.- 1.C Vector fields.- 1.C.1 Definitions.- 1.C.2 Another definition for the tangent space.- 1.C.3 Integral curves and flow of a vector field.- 1.C.4 Image of a vector field by a diffeomorphism.- 1.D Baby Lie groups.- 1.D.1 Definitions.- 1.D.2 Adjoint representation.- 1.E Covering maps and fibrations.- 1.E.1 Covering maps and quotients by a discrete group.- 1.E.2 Submersions and fibrations.- 1.E.3 Homogeneous spaces.- 1.F Tensors.- 1.F.1 Tensor product (a digest).- 1.F.2 Tensor bundles.- 1.F.3 Operations on tensors.- 1.F.4 Lie derivatives.- 1.F.5 Local operators, differential operators.- 1.F.6 A characterization for tensors.- 1.G. Differential forms.- 1.G.1 Definitions.- 1.G.2 Exterior derivative.- 1.G.3 Volume forms.- 1.G.4 Integration on an oriented manifold.- 1.G.5 Haar measure on a Lie group.- 1.H Partitions of unity.- 2 Riemannian metrics.- 2.A Existence theorems and first examples.- 2.A.1 Basic definitions.- 2.A.2 Submanifolds of Euclidean or Minkowski spaces.- 2.A.3 Riemannian submanifolds, Riemannian products.- 2.A.4 Riemannian covering maps, flat tori.- 2.A.5 Riemannian submersions, complex projective space.- 2.A.6 Homogeneous Riemannian spaces.- 2.B Covariant derivative.- 2.B.1 Connections.- 2.B.2 Canonical connection of a Riemannian submanifold.- 2.B.3 Extension of the covariant derivative to tensors.- 2.B.4 Covariant derivative along a curve.- 2.B.5 Parallel transport.- 2.B.6 natural metric on the tangent bundle.- 2.C Geodesies.- 2.C.1 Definition, first examples.- 2.C.2 Local existence and uniqueness for geodesies, exponential map.- 2.C.3 Riemannian manifolds as metric spaces.- 2.C.4 An invitation to isosystolic inequalities.- 2.C.5 Complete Riemannian manifolds, Hopf-Rinow theorem.- 2.C.6 Geodesies and submersions, geodesies of PnC.- 2.C.7 Cut-locus.- 2.C.8 The geodesic flow.- 2.D A glance at pseudo-Riemannian manifolds.- 2.D.1 What remains true?.- 2.D.2 Space, time and light-like curves.- 2.D.3 Lorentzian analogs of Euclidean spaces, spheres and hegeode spaces.- 2.D.4 (In)completeness.- 2.D.5 The Schwarzschild model.- 2.D.6 Hyperbolicity versus ellipticity.- 3 Curvature.- 3.A. The curvature tensor.- 3.A.1 Second covariant derivative.- 3.A.2 Algebraic properties of the curvature tensor.- 3.A.3 Computation of curvature: some examples.- 3.A.4 Ricci curvature, scalar curvature.- 3.B. First and second variation.- 3.B.1 Technical preliminaries.- 3.B.2 First variation formula.- 3.B.3 Second variation formula.- 3.C. Jacobi vector fields.- 3.C.1 Basic topics about second derivatives.- 3.C.2 Index form.- 3.C.3 Jacobi fields and exponential map.- 3.C.4 Applications.- 3.D. Riemannian submersions and curvature.- 3.D.1 Riemannian submersions and connections.- 3.D.2 Jacobi fields of PnC.- 3.D.3 O’Neill’s formula.- 3.D.4 Curvature and length of small circles. Application to Riemannian submersions.- 3.E. The behavior of length and energy in the neighborhood of a geodesic.- 3.E.1 Gauss lemma.- 3.E.2 Conjugate points.- 3.E.3 Some properties of the cut-locus.- 3.F Manifolds with constant sectional curvature.- 3.G Topology and curvature: two basic results.- 3.G.1 Myers’ theorem.- 3.G.2 Cartan-Hadamard’s theorem.- 3.H. Curvature and volume.- 3.H.1 Densities on a differentiable manifold.- 3.H.2 Canonical measure of a Riemannian manifold.- 3.H.3 Examples: spheres, hyperbolic spaces, complex projective spaces.- 3.H.4 Small balls and scalar curvature.- 3.H.5 Volume estimates.- 3.I. Curvature and growth of the fundamental group.- 3.I.1 Growth of finite type groups.- 3.I.2 Growth of the fundamental group of compact manifolds with negative curvature.- 3.J. Curvature and topology: some important results.- 3.J.1 Integral formulas.- 3.J.2 (Geo)metric methods.- 3.J.3 Analytic methods.- 3.J.4 Coarse point of view: compactness theorems.- 3.K. Curvature tensors and representations of the orthogonal group.- 3.K.1 Decomposition of the space of curvature tensors.- 3.K.2 Conformally flat manifolds.- 3.K.3 The Second Bianchi identity.- 3.L. Hyperbolic geometry.- 3.L.1 Introduction.- 3.L.2 Angles and distances in the hyperbolic plane.- 3.L.3 Polygons with “many” right angles.- 3.L.4 Compact surfaces.- 3.L.5 Hyperbolic trigonometry.- 3.L.6 Prescribing constant negative curvature.- 3.L.7 A few words about higher dimension.- 3.M. Conformai geometry.- 3.M.2 Introduction.- 3.M.3 The Möbius group.- 3.M.4 Conformai, elliptic and hyperbolic geometry.- 4 Analysis on manifolds.- 4.A. Manifolds with boundary.- 4.A.1 Introduction.- 4.A.2 Stokes theorem and integration by parts.- 4.B. Bishop inequality.- 4.B.1 Some commutation formulas.- 4.B.2 Laplacian of the distance function.- 4.B.3 Another proof of Bishop’s inequality.- 4.B.4 Heintze-Karcher inequality.- 4.C. Differential forms and cohomology.- 4.C.1 The de Rham complex.- 4.C.2 Differential operators and their formal adjoints.- 4.C.3 The Hodge-de Rham theorem.- 4.C.4 A second visit to the Bochner method.- 4.D. Basic spectral geometry.- 4.D.1 The Laplace operator and the wave equation.- 4.D.2 Statement of basic results on the spectrum.- 4.E. Some examples of spectra.- 4.E.1 Introduction.- 4.E.2 The spectrum of flat tori.- 4.E.3 Spectrum of (Sn, can).- 4.F The minimax principle.- 4.G Eigenvalues estimates.- 4.G.1 Introduction.- 4.G.2 Bishop’s inequality and coarse estimates.- 4.G.3 Some consequences of Bishop’s theorem.- 4.G.4 Lower bounds for the first eigenvalue.- 4.H. Paul Levy’s isoperimetric inequality.- 4.H.1 The statement.- 4.H.2 The proof.- 5 Riemannian submanifolds.- 5.A. Curvature of submanifolds.- 5.A.1 Second fundamental form.- 5.A.2 Curvature of hypersurfaces.- 5.A.3 Application to explicit computations of curvatures.- 5.B Curvature and convexity.- 5.C Minimal surfaces.- 5.C.1 First results.- 5.C.2 Surfaces with constant mean curvature.- A Some extra problems.- B Solutions of exercises.- List of figures.
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