In succesion to former international meetings on differential geometry held in Hungary and also as a satellite conference of ECM96, the European Mathematical Congress, a Conference on Differential Geometry took place in Budapest from July 27 to July 30, 1996. The host of the Conference was Lorand Eotvos University. The Conference had the following Programme Committee: D.V. Alekseevsky, J.J. Duistermaat, J. Eells, A. Haefliger, O. Kowalski, S. Marchifava, J. Szenthe, L. Tamassy, L. Vanhecke. The participants came mainly from Europe and their total number was 190. The programme included plenary lectures by J. Eliashberg, S. Gallot, O. Kowalski, B. Leeb, and also 135 lectures in 4 sections. The social events, an opening reception and a farewel party, presented inspiring atmosphere to create scientific contacts and also for fruitful discussions. In preparation of the Conference and during it B. Csikos and G. Moussong were constanly ready to help. The present volume contains detailed versions of lectures presented at the Conference and also a list of participants. The subjects cover a wide variety of topics in differential geometry and its applications and all of them contain essential new developments in their respective subjects. It is my pleasant duty to thank the participants who contributed to the success of the Conference, especially those who offered us their manuscripts for publication and also the referees who made several important observa tions. The preparation of the volume was managed with the assistance of E. Daroczy-Kiss."

This monograph is based on the author's results on the Riemannian ge ometry of foliations with nonnegative mixed curvature and on the geometry of sub manifolds with generators (rulings) in a Riemannian space of nonnegative curvature. The main idea is that such foliated (sub) manifolds can be decom posed when the dimension of the leaves (generators) is large. The methods of investigation are mostly synthetic. The work is divided into two parts, consisting of seven chapters and three appendices. Appendix A was written jointly with V. Toponogov. Part 1 is devoted to the Riemannian geometry of foliations. In the first few sections of Chapter I we give a survey of the basic results on foliated smooth manifolds (Sections 1.1-1.3), and finish in Section 1.4 with a discussion of the key problem of this work: the role of Riemannian curvature in the study of foliations on manifolds and submanifolds."

This book provides an up-to-date presentation of homogeneous pseudo-Riemannian structures, an essential tool in the study of pseudo-Riemannian homogeneous spaces. Benefiting from large symmetry groups, these spaces are of high interest in Geometry and Theoretical Physics. Since the seminal book by Tricerri and Vanhecke, the theory of homogeneous structures has been considerably developed and many applications have been found. The present work covers a gap in the literature of more than 35 years, presenting the latest contributions to the field in a modern geometric approach, with special focus on manifolds equipped with pseudo-Riemannian metrics. This unique reference on the topic will be of interest to researchers working in areas of mathematics where homogeneous spaces play an important role, such as Differential Geometry, Global Analysis, General Relativity, and Particle Physics.

Through two previous editions, the third edition of this popular and intriguing text takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Requiring only multivariable calculus and linear algebra, it develops students' geometric intuition through interactive graphics applets. Applets are presented in Maple workbook format, which readers can access using the free Maple Player. The book explains the reasons for various definitions while the interactive applets offer motivation for definitions, allowing students to explore examples further, and give a visual explanation of complicated theorems. The ability to change parametric curves and parametrized surfaces in an applet lets students probe the concepts far beyond what static text permits. Investigative project ideas promote student research. At users of the previous editions' request, this third edition offers a broader list of exercises. More elementary exercises are added and some challenging problems are moved later in exercise sets to assure more graduated progress. The authors also add hints to motivate students grappling with the more difficult exercises. This student-friendly and readable approach offers additional examples, well-placed to assist student comprehension. In the presentation of the Gauss-Bonnet Theorem, the authors provide more intuition and stepping-stones to help students grasp phenomena behind it. Also, the concept of a homeomorphism is new to students even though it is a key theoretical component of the definition of a regular surface. Providing more examples show students how to prove certain functions are homeomorphisms.

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.

This self-containedtext is an excellent introductionto Lie groups and their actions on manifolds. Theauthors start withan elementarydiscussion of matrix groups, followed by chapters devoted to the basic structure and representation theory of finite dimensinal Lie algebras. They then turn to global issues, demonstrating the key issue of the interplay between differential geometry and Lie theory. Special emphasis is placed on homogeneous spaces and invariant geometric structures. The last section of the book is dedicated to the structure theory of Lie groups. Particularly, they focus on maximal compact subgroups, dense subgroups, complex structures, and linearity.

This text is accessible to a broad range of mathematicians and graduate students; it will be useful both as a graduate textbook and as a research reference."

This volume presents recent developments in geometric structures on Riemannian manifolds and their discretizations. With chapters written by recognized experts, these discussions focus on contact structures, Kahler structures, fiber bundle structures and Einstein metrics. It also contains works on the geometric approach on coding theory.For researchers and students, this volume forms an invaluable source to learn about these subjects that are not only in the field of differential geometry but also in other wide related areas. It promotes and deepens the study of geometric structures.

The book is devoted to recent research in the global variational theory on smooth manifolds. Its main objective is an extension of the classical variational calculus on Euclidean spaces to (topologically nontrivial) finite-dimensional smooth manifolds; to this purpose the methods of global analysis of differential forms are used. Emphasis is placed on the foundations of the theory of variational functionals on fibered manifolds - relevant geometric structures for variational principles in geometry, physical field theory and higher-order fibered mechanics. The book chapters include: - foundations of jet bundles and analysis of differential forms and vector fields on jet bundles, - the theory of higher-order integral variational functionals for sections of a fibred space, the (global) first variational formula in infinitesimal and integral forms- extremal conditions and the discussion of Noether symmetries and generalizations,- the inverse problems of the calculus of variations of Helmholtz type- variational sequence theory and its consequences for the global inverse problem (cohomology conditions)- examples of variational functionals of mathematical physics. Complete formulations and proofs of all basic assertions are given, based on theorems of global analysis explained in the Appendix.

The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks:- Geometry and topology of fibre bundles,- Clifford algebras, spin structures and Dirac operators,- Gauge theory.Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory.The first building block includes a number of specific topics, like invariant connections, universal connections, H-structures and the Postnikov approximation of classifying spaces.Given the great importance of Dirac operators in gauge theory, a complete proof of the Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional reduction) and the investigation of the structure of the gauge orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem, anomalies and the implementation of the non-generic gauge orbit strata in the framework of Hamiltonian lattice gauge theory.The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from a graduate level.

This is a volume originating from the Conference on Partial Differential Equations and Applications, which was held in Moscow in November 2018 in memory of professor Boris Sternin and attracted more than a hundred participants from eighteen countries. The conference was mainly dedicated to partial differential equations on manifolds and their applications in mathematical physics, geometry, topology, and complex analysis. The volume contains selected contributions by leading experts in these fields and presents the current state of the art in several areas of PDE. It will be of interest to researchers and graduate students specializing in partial differential equations, mathematical physics, topology, geometry, and their applications. The readers will benefit from the interplay between these various areas of mathematics.

This 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 theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Differential geometry is the study of the curvature and calculus of curves and surfaces. "A New Approach to Differential Geometry using Clifford's Geometric Algebra" simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.

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Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

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This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. It is written by three leading experts on the interplay between two important fields — singularity theory and differential geometry. The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the objects of attention to singularities of plane curves and surfaces in the Euclidean 3-space. In particular, by presenting the singular curvature, which originated through research by the authors, the Gauss–Bonnet theorem for surfaces is generalized to those with singularities. The Gauss–Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. The book also elucidates the notion of Riemannian manifolds with singularities. These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material. Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful guide for students and researchers interested in this subject.

The textbook is a very good start into the mathematical field of topology. A variety of topological concepts with some elementary applications are introduced. It is organized in such a way that the reader gets to significant applications quickly.This revised version corrects the many discrepancies in the earlier edition. The emphasis is on the geometric understanding and the use of new concepts, indicating that topology is really the language of modern mathematics.

This book develops a novel approach to perturbative quantum field theory: starting with a perturbative formulation of classical field theory, quantization is achieved by means of deformation quantization of the underlying free theory and by applying the principle that as much of the classical structure as possible should be maintained. The resulting formulation of perturbative quantum field theory is a version of the Epstein-Glaser renormalization that is conceptually clear, mathematically rigorous and pragmatically useful for physicists. The connection to traditional formulations of perturbative quantum field theory is also elaborated on, and the formalism is illustrated in a wealth of examples and exercises.

This work is at the crossroads of a number of mathematical areas, including algebraic geometry, several complex variables, differential geometry, and representation theory. It is the first book to cover complex tori, among the simplest of complex manifolds, which are important to research in the above areas. The book gives a systematic approach to the theory, presents new results, and includes an up-to-date bibliography.

This 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.

This book deals with the classical theory of Nevanlinna on the value distribution of meromorphic functions of one complex variable, based on minimum prerequisites for complex manifolds. The theory was extended to several variables by S. Kobayashi, T. Ochiai, J. Carleson, and P. Griffiths in the early 1970s. K. Kodaira took up this subject in his course at The University of Tokyo in 1973 and gave an introductory account of this development in the context of his final paper, contained in this book. The first three chapters are devoted to holomorphic mappings from C to complex manifolds. In the fourth chapter, holomorphic mappings between higher dimensional manifolds are covered. The book is a valuable treatise on the Nevanlinna theory, of special interests to those who want to understand Kodaira's unique approach to basic questions on complex manifolds.

This monograph presents in a unified manner the use of the Morse index, and especially its connections to the maximum principle, in the study of nonlinear elliptic equations. The knowledge or a bound on the Morse index of a solution is a very important qualitative information which can be used in several ways for different problems, in order to derive uniqueness, existence or nonexistence, symmetry, and other properties of solutions.

This 4-th edition of the leading reference volume on distance metrics is characterized by updated and rewritten sections on some items suggested by experts and readers, as well a general streamlining of content and the addition of essential new topics. Though the structure remains unchanged, the new edition also explores recent advances in the use of distances and metrics for e.g. generalized distances, probability theory, graph theory, coding theory, data analysis. New topics in the purely mathematical sections include e.g. the Vitanyi multiset-metric, algebraic point-conic distance, triangular ratio metric, Rossi-Hamming metric, Taneja distance, spectral semimetric between graphs, channel metrization, and Maryland bridge distance. The multidisciplinary sections have also been supplemented with new topics, including: dynamic time wrapping distance, memory distance, allometry, atmospheric depth, elliptic orbit distance, VLBI distance measurements, the astronomical system of units, and walkability distance. Leaving aside the practical questions that arise during the selection of a 'good' distance function, this work focuses on providing the research community with an invaluable comprehensive listing of the main available distances. As well as providing standalone introductions and definitions, the encyclopedia facilitates swift cross-referencing with easily navigable bold-faced textual links to core entries. In addition to distances themselves, the authors have collated numerous fascinating curiosities in their Who's Who of metrics, including distance-related notions and paradigms that enable applied mathematicians in other sectors to deploy research tools that non-specialists justly view as arcane. In expanding access to these techniques, and in many cases enriching the context of distances themselves, this peerless volume is certain to stimulate fresh research.

This volume collects papers based on talks given at the conference "Geometrias'19: Polyhedra and Beyond", held in the Faculty of Sciences of the University of Porto between September 5-7, 2019 in Portugal. These papers explore the conference's theme from an interdisciplinary standpoint, all the while emphasizing the relevance of polyhedral geometry in contemporary academic research and professional practice. They also investigate how this topic connects to mathematics, art, architecture, computer science, and the science of representation. Polyhedra and Beyond will help inspire scholars, researchers, professionals, and students of any of these disciplines to develop a more thorough understanding of polyhedra.

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations. Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds", the focus has now changed from the general theory of manifolds to general differential geometry, and includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, a generalization of the Cartan-Hadamard theorem, the semiparallelogram law of Bruhat-Tits and its equivalence with seminegative curvature and the exponential map distance increasing property, a major example of seminegative curvature (the space of positive definite symmetric real matrices), automorphisms and symmetries, and immersions and submersions. These are all covered for infinite-dimensional manifolds, modeled on Banach and Hilbert Spaces, at no cost in complications, and some gain in the elegance of the proofs. In the finite-dimensional case, differential forms of top degree are discussed, leading to Stokes' theorem (even for manifolds with singular boundary), and several of its applications to the differential or Riemannian case. Basic formulas concerning the Laplacian are given, exhibiting several of its features in immersions and submersions.

Two surveys introducing readers to the subjects of harmonic analysis on semi-simple spaces and group theoretical methods, and preparing them for the study of more specialised literature. This book will be very useful to students and researchers in mathematics, theoretical physics and those chemists dealing with quantum systems.

This book includes selected papers presented at the MIMS (Mediterranean Institute for the Mathematical Sciences) - GGTM (Geometry and Topology Grouping for the Maghreb) conference, held in memory of Mohammed Salah Baouendi, a most renowned figure in the field of several complex variables, who passed away in 2011. All research articles were written by leading experts, some of whom are prize winners in the fields of complex geometry, algebraic geometry and analysis. The book offers a valuable resource for all researchers interested in recent developments in analysis and geometry.