Introduction To Geometry Of Manifolds With Symmetry

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Introduction to Geometry of Manifolds with Symmetry

V.V. Trofimov
Introduction to Geometry of Manifolds with Symmetry Book PDF

Author : V.V. Trofimov
Publisher : Springer Science & Business Media
Page : 339 pages
File Size : 44,9 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401719612

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Introduction to Geometry of Manifolds with Symmetry by V.V. Trofimov Pdf

One ofthe most important features of the development of physical and mathematical sciences in the beginning of the 20th century was the demolition of prevailing views of the three-dimensional Euclidean space as the only possible mathematical description of real physical space. Apriorization of geometrical notions and identification of physical 3 space with its mathematical modellR were characteristic for these views. The discovery of non-Euclidean geometries led mathematicians to the understanding that Euclidean geometry is nothing more than one of many logically admissible geometrical systems. Relativity theory amended our understanding of the problem of space by amalgamating space and time into an integral four-dimensional manifold. One of the most important problems, lying at the crossroad of natural sciences and philosophy is the problem of the structure of the world as a whole. There are a lot of possibilities for the topology offour dimensional space-time, and at first sight a lot of possibilities arise in cosmology. In principle, not only can the global topology of the universe be complicated, but also smaller scale topological structures can be very nontrivial. One can imagine two "usual" spaces connected with a "throat", making the topology of the union complicated.

The Geometry of Walker Manifolds

Miguel Brozos-Vázquez
The Geometry of Walker Manifolds Book PDF

Author : Miguel Brozos-Vázquez
Publisher : Morgan & Claypool Publishers
Page : 178 pages
File Size : 46,5 Mb
Release : 2009
Category : Mathematics
ISBN : 9781598298192

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The Geometry of Walker Manifolds by Miguel Brozos-Vázquez Pdf

Basic algebraic notions -- Introduction -- A historical perspective in the algebraic context -- Algebraic preliminaries -- Jordan normal form -- Indefinite geometry -- Algebraic curvature tensors -- Hermitian and para-Hermitian geometry -- The Jacobi and skew symmetric curvature operators -- Sectional, Ricci, scalar, and Weyl curvature -- Curvature decompositions -- Self-duality and anti-self-duality conditions -- Spectral geometry of the curvature operator -- Osserman and conformally Osserman models -- Osserman curvature models in signature (2, 2) -- Ivanov-Petrova curvature models -- Osserman Ivanov-Petrova curvature models -- Commuting curvature models -- Basic geometrical notions -- Introduction -- History -- Basic manifold theory -- The tangent bundle, lie bracket, and lie groups -- The cotangent bundle and symplectic geometry -- Connections, curvature, geodesics, and holonomy -- Pseudo-Riemannian geometry -- The Levi-Civita connection -- Associated natural operators -- Weyl scalar invariants -- Null distributions -- Pseudo-Riemannian holonomy -- Other geometric structures -- Pseudo-Hermitian and para-Hermitian structures -- Hyper-para-Hermitian structures -- Geometric realizations -- Homogeneous spaces, and curvature homogeneity -- Technical results in differential equations -- Walker structures -- Introduction -- Historical development -- Walker coordinates -- Examples of Walker manifolds -- Hypersurfaces with nilpotent shape operators -- Locally conformally flat metrics with nilpotent Ricci operator -- Degenerate pseudo-Riemannian homogeneous structures -- Para-Kaehler geometry -- Two-step nilpotent lie groups with degenerate center -- Conformally symmetric pseudo-Riemannian metrics -- Riemannian extensions -- The affine category -- Twisted Riemannian extensions defined by flat connections -- Modified Riemannian extensions defined by flat connections -- Nilpotent Walker manifolds -- Osserman Riemannian extensions -- Ivanov-Petrova Riemannian extensions -- Three-dimensional Lorentzian Walker manifolds -- Introduction -- History -- Three dimensional Walker geometry -- Adapted coordinates -- The Jordan normal form of the Ricci operator -- Christoffel symbols, curvature, and the Ricci tensor -- Locally symmetric Walker manifolds -- Einstein-like manifolds -- The spectral geometry of the curvature tensor -- Curvature commutativity properties -- Local geometry of Walker manifolds with -- Foliated Walker manifolds -- Contact Walker manifolds -- Strict Walker manifolds -- Three dimensional homogeneous Lorentzian manifolds -- Three dimensional lie groups and lie algebras -- Curvature homogeneous Lorentzian manifolds -- Diagonalizable Ricci operator -- Type II Ricci operator -- Four-dimensional Walker manifolds -- Introduction -- History -- Four-dimensional Walker manifolds -- Almost para-Hermitian geometry -- Isotropic almost para-Hermitian structures -- Characteristic classes -- Self-dual Walker manifolds -- The spectral geometry of the curvature tensor -- Introduction -- History -- Four-dimensional Osserman metrics -- Osserman metrics with diagonalizable Jacobi operator -- Osserman Walker type II metrics -- Osserman and Ivanov-Petrova metrics -- Riemannian extensions of affine surfaces -- Affine surfaces with skew symmetric Ricci tensor -- Affine surfaces with symmetric and degenerate Ricci tensor -- Riemannian extensions with commuting curvature operators -- Other examples with commuting curvature operators -- Hermitian geometry -- Introduction -- History -- Almost Hermitian geometry of Walker manifolds -- The proper almost Hermitian structure of a Walker manifold -- Proper almost hyper-para-Hermitian structures -- Hermitian Walker manifolds of dimension four -- Proper Hermitian Walker structures -- Locally conformally Kaehler structures -- Almost Kaehler Walker four-dimensional manifolds -- Special Walker manifolds -- Introduction -- History -- Curvature commuting conditions -- Curvature homogeneous strict Walker manifolds -- Bibliography.

Geometry of Manifolds with Non-negative Sectional Curvature

Owen Dearricott,Fernando Galaz-García,Lee Kennard,Catherine Searle,Gregor Weingart,Wolfgang Ziller
Geometry of Manifolds with Non negative Sectional Curvature Book PDF

Author : Owen Dearricott,Fernando Galaz-García,Lee Kennard,Catherine Searle,Gregor Weingart,Wolfgang Ziller
Publisher : Springer
Page : 196 pages
File Size : 51,8 Mb
Release : 2014-07-22
Category : Mathematics
ISBN : 9783319063737

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Geometry of Manifolds with Non-negative Sectional Curvature by Owen Dearricott,Fernando Galaz-García,Lee Kennard,Catherine Searle,Gregor Weingart,Wolfgang Ziller Pdf

Providing an up-to-date overview of the geometry of manifolds with non-negative sectional curvature, this volume gives a detailed account of the most recent research in the area. The lectures cover a wide range of topics such as general isometric group actions, circle actions on positively curved four manifolds, cohomogeneity one actions on Alexandrov spaces, isometric torus actions on Riemannian manifolds of maximal symmetry rank, n-Sasakian manifolds, isoparametric hypersurfaces in spheres, contact CR and CR submanifolds, Riemannian submersions and the Hopf conjecture with symmetry. Also included is an introduction to the theory of exterior differential systems.

An Introduction to Manifolds

Loring W. Tu
An Introduction to Manifolds Book PDF

Author : Loring W. Tu
Publisher : Springer Science & Business Media
Page : 426 pages
File Size : 51,7 Mb
Release : 2010-10-05
Category : Mathematics
ISBN : 9781441974006

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An Introduction to Manifolds by Loring W. Tu Pdf

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised

William Munger Boothby
An Introduction to Differentiable Manifolds and Riemannian Geometry Revised Book PDF

Author : William Munger Boothby
Publisher : Gulf Professional Publishing
Page : 444 pages
File Size : 54,6 Mb
Release : 2003
Category : Mathematics
ISBN : 0121160513

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An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised by William Munger Boothby Pdf

The second edition of An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject. Line and surface integrals Divergence and curl of vector fields

Differential Geometry and Symmetric Spaces

Sigurdur Helgason
Differential Geometry and Symmetric Spaces Book PDF

Author : Sigurdur Helgason
Publisher : American Mathematical Society
Page : 504 pages
File Size : 42,8 Mb
Release : 2024-04-05
Category : Mathematics
ISBN : 9781470476878

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Differential Geometry and Symmetric Spaces by Sigurdur Helgason Pdf

Sigurdur Helgason's Differential Geometry and Symmetric Spaces was quickly recognized as a remarkable and important book. For many years, it was the standard text both for Riemannian geometry and for the analysis and geometry of symmetric spaces. Several generations of mathematicians relied on it for its clarity and careful attention to detail. Although much has happened in the field since the publication of this book, as demonstrated by Helgason's own three-volume expansion of the original work, this single volume is still an excellent overview of the subjects. For instance, even though there are now many competing texts, the chapters on differential geometry and Lie groups continue to be among the best treatments of the subjects available. There is also a well-developed treatment of Cartan's classification and structure theory of symmetric spaces. The last chapter, on functions on symmetric spaces, remains an excellent introduction to the study of spherical functions, the theory of invariant differential operators, and other topics in harmonic analysis. This text is rightly called a classic.

The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds

Peter B. Gilkey
The Geometry of Curvature Homogeneous Pseudo Riemannian Manifolds Book PDF

Author : Peter B. Gilkey
Publisher : World Scientific
Page : 389 pages
File Size : 42,7 Mb
Release : 2007
Category : Science
ISBN : 9781860947858

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The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds by Peter B. Gilkey Pdf

"Pseudo-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."--BOOK JACKET.

Riemannian Manifolds

John M. Lee
Riemannian Manifolds Book PDF

Author : John M. Lee
Publisher : Springer Science & Business Media
Page : 232 pages
File Size : 44,9 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387227269

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Riemannian Manifolds by John M. Lee Pdf

This 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.

Complex, Contact and Symmetric Manifolds

Oldrich Kowalski,Emilio E. Musso,Domenico Perrone
Complex Contact and Symmetric Manifolds Book PDF

Author : Oldrich Kowalski,Emilio E. Musso,Domenico Perrone
Publisher : Birkhäuser
Page : 278 pages
File Size : 51,8 Mb
Release : 2008-11-01
Category : Mathematics
ISBN : 081767053X

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Complex, Contact and Symmetric Manifolds by Oldrich Kowalski,Emilio E. Musso,Domenico Perrone Pdf

* Contains research and survey articles by well known and respected mathematicians on recent developments and research trends in differential geometry and topology * Dedicated in honor of Lieven Vanhecke, as a tribute to his many fruitful and inspiring contributions to these fields * Papers include all necessary introductory and contextual material to appeal to non-specialists, as well as researchers and differential geometers

Geometry of Nonpositively Curved Manifolds

Patrick Eberlein
Geometry of Nonpositively Curved Manifolds Book PDF

Author : Patrick Eberlein
Publisher : University of Chicago Press
Page : 460 pages
File Size : 47,7 Mb
Release : 1996
Category : Mathematics
ISBN : 0226181987

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Geometry of Nonpositively Curved Manifolds by Patrick Eberlein Pdf

Starting from the foundations, the author presents an almost entirely self-contained treatment of differentiable spaces of nonpositive curvature, focusing on the symmetric spaces in which every geodesic lies in a flat Euclidean space of dimension at least two. The book builds to a discussion of the Mostow Rigidity Theorem and its generalizations, and concludes by exploring the relationship in nonpositively curved spaces between geometric and algebraic properties of the fundamental group. This introduction to the geometry of symmetric spaces of non-compact type will serve as an excellent guide for graduate students new to the material, and will also be a useful reference text for mathematicians already familiar with the subject.

Differential Geometry of Manifolds

Stephen Lovett
Differential Geometry of Manifolds Book PDF

Author : Stephen Lovett
Publisher : CRC Press
Page : 430 pages
File Size : 43,9 Mb
Release : 2010-06-11
Category : Mathematics
ISBN : 9781439865460

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Differential Geometry of Manifolds by Stephen Lovett Pdf

From the coauthor of Differential Geometry of Curves and Surfaces, this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices

Classical Mirror Symmetry

Masao Jinzenji
Classical Mirror Symmetry Book PDF

Author : Masao Jinzenji
Publisher : Springer
Page : 140 pages
File Size : 49,7 Mb
Release : 2018-04-18
Category : Science
ISBN : 9789811300561

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Classical Mirror Symmetry by Masao Jinzenji Pdf

This book furnishes a brief introduction to classical mirror symmetry, a term that denotes the process of computing Gromov–Witten invariants of a Calabi–Yau threefold by using the Picard–Fuchs differential equation of period integrals of its mirror Calabi–Yau threefold. The book concentrates on the best-known example, the quintic hypersurface in 4-dimensional projective space, and its mirror manifold.First, there is a brief review of the process of discovery of mirror symmetry and the striking result proposed in the celebrated paper by Candelas and his collaborators. Next, some elementary results of complex manifolds and Chern classes needed for study of mirror symmetry are explained. Then the topological sigma models, the A-model and the B-model, are introduced. The classical mirror symmetry hypothesis is explained as the equivalence between the correlation function of the A-model of a quintic hyper-surface and that of the B-model of its mirror manifold.On the B-model side, the process of construction of a pair of mirror Calabi–Yau threefold using toric geometry is briefly explained. Also given are detailed explanations of the derivation of the Picard–Fuchs differential equation of the period integrals and on the process of deriving the instanton expansion of the A-model Yukawa coupling based on the mirror symmetry hypothesis.On the A-model side, the moduli space of degree d quasimaps from CP^1 with two marked points to CP^4 is introduced, with reconstruction of the period integrals used in the B-model side as generating functions of the intersection numbers of the moduli space. Lastly, a mathematical justification for the process of the B-model computation from the point of view of the geometry of the moduli space of quasimaps is given.The style of description is between that of mathematics and physics, with the assumption that readers have standard graduate student backgrounds in both disciplines.

Differential Geometry

Loring W. Tu
Differential Geometry Book PDF

Author : Loring W. Tu
Publisher : Springer
Page : 347 pages
File Size : 53,7 Mb
Release : 2017-06-01
Category : Mathematics
ISBN : 9783319550848

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Differential Geometry by Loring W. Tu Pdf

This 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.

Differential Geometry of Singular Spaces and Reduction of Symmetry

Jędrzej Śniatycki
Differential Geometry of Singular Spaces and Reduction of Symmetry Book PDF

Author : Jędrzej Śniatycki
Publisher : Cambridge University Press
Page : 249 pages
File Size : 41,9 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9781107022713

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Differential Geometry of Singular Spaces and Reduction of Symmetry by Jędrzej Śniatycki Pdf

A complete presentation of the theory of differential spaces, with applications to the study of singularities in systems with symmetry.

Lectures on Symplectic Geometry

Ana Cannas da Silva
Lectures on Symplectic Geometry Book PDF

Author : Ana Cannas da Silva
Publisher : Springer
Page : 220 pages
File Size : 40,6 Mb
Release : 2004-10-27
Category : Mathematics
ISBN : 9783540453307

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Lectures on Symplectic Geometry by Ana Cannas da Silva Pdf

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.