Cartan Geometries And Their Symmetries

Author : Mike Crampin,David Saunders
Publisher : Springer
Page : 290 pages
File Size : 43,5 Mb
Release : 2016-05-20
Category : Mathematics
ISBN : 9789462391925

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Cartan Geometries and their Symmetries by Mike Crampin,David Saunders Pdf

In this book we first review the ideas of Lie groupoid and Lie algebroid, and the associated concepts of connection. We next consider Lie groupoids of fibre morphisms of a fibre bundle, and the connections on such groupoids together with their symmetries. We also see how the infinitesimal approach, using Lie algebroids rather than Lie groupoids, and in particular using Lie algebroids of vector fields along the projection of the fibre bundle, may be of benefit. We then introduce Cartan geometries, together with a number of tools we shall use to study them. We take, as particular examples, the four classical types of geometry: affine, projective, Riemannian and conformal geometry. We also see how our approach can start to fit into a more general theory. Finally, we specialize to the geometries (affine and projective) associated with path spaces and geodesics, and consider their symmetries and other properties.

Author : Ercüment H. Ortaçgil
Publisher : Oxford University Press
Page : 240 pages
File Size : 49,5 Mb
Release : 2018-06-28
Category : Mathematics
ISBN : 9780192554840

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An Alternative Approach to Lie Groups and Geometric Structures by Ercüment H. Ortaçgil Pdf

This book presents a new and innovative approach to Lie groups and differential geometry. Rather than compiling and reviewing the existing material on this classical subject, Professor Ortaçgil instead questions the foundations of the subject, and proposes a new direction. Aimed at the curious and courageous mathematician, this book aims to provoke further debate and inspire further development of this original research.

Author : Simon Gindikin
Publisher : Springer Science & Business Media
Page : 396 pages
File Size : 50,8 Mb
Release : 1996-06-27
Category : Mathematics
ISBN : 0817638288

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Topics in Geometry by Simon Gindikin Pdf

This collection of articles serves to commemorate the legacy of Joseph D'Atri, who passed away on April 29, 1993, a few days after his 55th birthday. Joe D' Atri is credited with several fundamental discoveries in ge ometry. In the beginning of his mathematical career, Joe was interested in the generalization of symmetrical spaces in the E. Cart an sense. Symmetric spaces, differentiated from other homogeneous manifolds by their geomet rical richness, allows the development of a deep analysis. Geometers have been constantly interested and challenged by the problem of extending the class of symmetric spaces so as to preserve their geometrical and analytical abundance. The name of D'Atri is tied to one of the most successful gen eralizations: Riemann manifolds in which (local) geodesic symmetries are volume-preserving (up to sign). In time, it turned out that the majority of interesting generalizations of symmetrical spaces are D'Atri spaces: natu ral reductive homogeneous spaces, Riemann manifolds whose geodesics are orbits of one-parameter subgroups, etc. The central place in D'Atri's research is occupied by homogeneous bounded domains in en, which are not symmetric. Such domains were discovered by Piatetskii-Shapiro in 1959, and given Joe's strong interest in the generalization of symmetric spaces, it was very natural for him to direct his research along this path.

Author : R.W. Sharpe
Publisher : Springer Science & Business Media
Page : 452 pages
File Size : 52,9 Mb
Release : 2000-11-21
Category : Mathematics
ISBN : 0387947329

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Differential Geometry by R.W. Sharpe Pdf

Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry.

Author : M. Crampin,F. A. E. Pirani
Publisher : Cambridge University Press
Page : 408 pages
File Size : 47,8 Mb
Release : 1986
Category : Mathematics
ISBN : 0521231906

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Applicable Differential Geometry by M. Crampin,F. A. E. Pirani Pdf

An introduction to geometrical topics used in applied mathematics and theoretical physics.

Author : D.L. Johnson
Publisher : Springer Science & Business Media
Page : 199 pages
File Size : 51,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447102434

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Symmetries by D.L. Johnson Pdf

" ... many eminent scholars, endowed with great geometric talent, make a point of never disclosing the simple and direct ideas that guided them, subordinating their elegant results to abstract general theories which often have no application outside the particular case in question. Geometry was becoming a study of algebraic, differential or partial differential equations, thus losing all the charm that comes from its being an art." H. Lebesgue, Ler;ons sur les Constructions Geometriques, Gauthier Villars, Paris, 1949. This book is based on lecture courses given to final-year students at the Uni versity of Nottingham and to M.Sc. students at the University of the West Indies in an attempt to reverse the process of expurgation of the geometry component from the mathematics curricula of universities. This erosion is in sharp contrast to the situation in research mathematics, where the ideas and methods of geometry enjoy ever-increasing influence and importance. In the other direction, more modern ideas have made a forceful and beneficial impact on the geometry of the ancients in many areas. Thus trigonometry has vastly clarified our concept of angle, calculus has revolutionised the study of plane curves, and group theory has become the language of symmetry.

Author : Oldřich Kowalski,Olga Krupkova
Publisher : World Scientific
Page : 732 pages
File Size : 43,6 Mb
Release : 2008
Category : Mathematics
ISBN : 9789812790613

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Differential Geometry and Its Applications by Oldřich Kowalski,Olga Krupkova Pdf

This volume contains invited lectures and selected research papers in the fields of classical and modern differential geometry, global analysis, and geometric methods in physics, presented at the 10th International Conference on Differential Geometry and its Applications (DGA2007), held in Olomouc, Czech Republic.The book covers recent developments and the latest results in the following fields: Riemannian geometry, connections, jets, differential invariants, the calculus of variations on manifolds, differential equations, Finsler structures, and geometric methods in physics. It is also a celebration of the 300th anniversary of the birth of one of the greatest mathematicians, Leonhard Euler, and includes the Euler lecture OC Leonhard Euler OCo 300 years onOCO by R Wilson. Notable contributors include J F Cariena, M Castrilln Lpez, J Erichhorn, J-H Eschenburg, I KoliO, A P Kopylov, J Korbai, O Kowalski, B Kruglikov, D Krupka, O Krupkovi, R L(r)andre, Haizhong Li, S Maeda, M A Malakhaltsev, O I Mokhov, J Muoz Masqu(r), S Preston, V Rovenski, D J Saunders, M Sekizawa, J Slovik, J Szilasi, L Tamissy, P Walczak, and others."

Author : Boris Kruglikov,Valentin Lychagin,Eldar Straume
Publisher : Springer Science & Business Media
Page : 394 pages
File Size : 46,7 Mb
Release : 2009-07-24
Category : Mathematics
ISBN : 9783642008733

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Differential Equations - Geometry, Symmetries and Integrability by Boris Kruglikov,Valentin Lychagin,Eldar Straume Pdf

The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories. This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 42,7 Mb
Release : 2024-07-03
Category : Electronic
ISBN : 9789814471947

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Differential Geometry and Its Applications by Anonim Pdf

Author : Thomas A. Ivey,Joseph M. Landsberg
Publisher : American Mathematical Soc.
Page : 455 pages
File Size : 43,8 Mb
Release : 2016-12-15
Category : Exterior differential systems
ISBN : 9781470409869

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Cartan for Beginners by Thomas A. Ivey,Joseph M. Landsberg Pdf

Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.

Author : Richard Montgomery
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 41,7 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821841655

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A Tour of Subriemannian Geometries, Their Geodesics and Applications by Richard Montgomery Pdf

Subriemannian geometries can be viewed as limits of Riemannian geometries. They arise naturally in many areas of pure (algebra, geometry, analysis) and applied (mechanics, control theory, mathematical physics) mathematics, as well as in applications (e.g., robotics). This book is devoted to the study of subriemannian geometries, their geodesics, and their applications. It starts with the simplest nontrivial example of a subriemannian geometry: the two-dimensional isoperimetric problem reformulated as a problem of finding subriemannian geodesics. Among topics discussed in other chapters of the first part of the book are an elementary exposition of Gromov's idea to use subriemannian geometry for proving a theorem in discrete group theory and Cartan's method of equivalence applied to the problem of understanding invariants of distributions. The second part of the book is devoted to applications of subriemannian geometry. In particular, the author describes in detail Berry's phase in quantum mechanics, the problem of a falling cat righting herself, that of a microorganism swimming, and a phase problem arising in the $N$-body problem. He shows that all these problems can be studied using the same underlying type of subriemannian geometry. The reader is assumed to have an introductory knowledge of differential geometry. This book that also has a chapter devoted to open problems can serve as a good introduction to this new, exciting area of mathematics.

Author : J. Śniatycki
Publisher : Cambridge University Press
Page : 249 pages
File Size : 49,7 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9781107067387

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Differential Geometry of Singular Spaces and Reduction of Symmetry by J. Śniatycki Pdf

In this book the author illustrates the power of the theory of subcartesian differential spaces for investigating spaces with singularities. Part I gives a detailed and comprehensive presentation of the theory of differential spaces, including integration of distributions on subcartesian spaces and the structure of stratified spaces. Part II presents an effective approach to the reduction of symmetries. Concrete applications covered in the text include reduction of symmetries of Hamiltonian systems, non-holonomically constrained systems, Dirac structures, and the commutation of quantization with reduction for a proper action of the symmetry group. With each application the author provides an introduction to the field in which relevant problems occur. This book will appeal to researchers and graduate students in mathematics and engineering.

Author : Michael Eastwood,Willard Miller
Publisher : Springer Science & Business Media
Page : 565 pages
File Size : 45,9 Mb
Release : 2009-04-23
Category : Mathematics
ISBN : 9780387738314

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Symmetries and Overdetermined Systems of Partial Differential Equations by Michael Eastwood,Willard Miller Pdf

This three-week summer program considered the symmetries preserving various natural geometric structures. There are two parts to the proceedings. The articles in the first part are expository but all contain significant new material. The articles in the second part are concerned with original research. All articles were thoroughly refereed and the range of interrelated work ensures that this will be an extremely useful collection.

Author : Elie Cartan,Serge? Pavlovich Finikov
Publisher : World Scientific
Page : 284 pages
File Size : 43,7 Mb
Release : 2001
Category : Mathematics
ISBN : 9810247478

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Riemannian Geometry in an Orthogonal Frame by Elie Cartan,Serge? Pavlovich Finikov Pdf

Elie Cartan's book Geometry of Riemannian Manifolds (1928) was one of the best introductions to his methods. It was based on lectures given by the author at the Sorbonne in the academic year 1925-26. A modernized and extensively augmented edition appeared in 1946 (2nd printing, 1951, and 3rd printing, 1988). Cartan's lectures in 1926-27 were different -- he introduced exterior forms at the very beginning and used extensively orthonormal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. The lectures were translated into Russian in the book Riemannian Geometry in an Orthogonal Frame (1960). This book has many innovations, such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature, an affine connection defined in a normal fiber bundle of a submanifold, etc. The only book of Elie Cartan that was not available in English, it has now been translated into English by Vladislav V Goldberg, the editor of the Russian edition.

Author : M Blagojevic
Publisher : CRC Press
Page : 522 pages
File Size : 54,7 Mb
Release : 2001-10-25
Category : Science
ISBN : 9781420034264

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Gravitation and Gauge Symmetries by M Blagojevic Pdf

In the course of the development of electromagnetic, weak and strong interactions, the concept of (internal) gauge invariance grew up and established itself as an unavoidable dynamical principle in particle physics. It is less known that the principle of equivalence, and the basic dynamical properties of the gravitational interaction can also be ex