Homogeneous Structures On Riemannian Manifolds

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Homogeneous Structures on Riemannian Manifolds

F. Tricerri,L. Vanhecke
Homogeneous Structures on Riemannian Manifolds Book PDF

Author : F. Tricerri,L. Vanhecke
Publisher : Cambridge University Press
Page : 145 pages
File Size : 54,6 Mb
Release : 1983-06-23
Category : Mathematics
ISBN : 9780521274890

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Homogeneous Structures on Riemannian Manifolds by F. Tricerri,L. Vanhecke Pdf

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

Homogeneous Structures on Riemannian Manifolds

Franco Tricerri,G Tricerri,L. Vanhecke
Homogeneous Structures on Riemannian Manifolds Book PDF

Author : Franco Tricerri,G Tricerri,L. Vanhecke
Publisher : Unknown
Page : 144 pages
File Size : 54,7 Mb
Release : 2014-05-14
Category : MATHEMATICS
ISBN : 1107087309

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Homogeneous Structures on Riemannian Manifolds by Franco Tricerri,G Tricerri,L. Vanhecke Pdf

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

Homogeneous Structures on Riemannian Manifolds

F. Tricerri,L. Vanhecke
Homogeneous Structures on Riemannian Manifolds Book PDF

Author : F. Tricerri,L. Vanhecke
Publisher : Cambridge University Press
Page : 144 pages
File Size : 44,7 Mb
Release : 1983-06-23
Category : Mathematics
ISBN : 0521274893

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Homogeneous Structures on Riemannian Manifolds by F. Tricerri,L. Vanhecke Pdf

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

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 : 45,7 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.

Pseudo-Riemannian Homogeneous Structures

Giovanni Calvaruso,Marco Castrillón López
Pseudo Riemannian Homogeneous Structures Book PDF

Author : Giovanni Calvaruso,Marco Castrillón López
Publisher : Springer
Page : 230 pages
File Size : 42,5 Mb
Release : 2019-08-14
Category : Mathematics
ISBN : 9783030181529

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Pseudo-Riemannian Homogeneous Structures by Giovanni Calvaruso,Marco Castrillón López Pdf

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.

Riemannian Manifolds and Homogeneous Geodesics

Valerii Berestovskii,Yurii Nikonorov
Riemannian Manifolds and Homogeneous Geodesics Book PDF

Author : Valerii Berestovskii,Yurii Nikonorov
Publisher : Springer Nature
Page : 482 pages
File Size : 53,9 Mb
Release : 2020-11-05
Category : Mathematics
ISBN : 9783030566586

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Riemannian Manifolds and Homogeneous Geodesics by Valerii Berestovskii,Yurii Nikonorov Pdf

This book is devoted to Killing vector fields and the one-parameter isometry groups of Riemannian manifolds generated by them. It also provides a detailed introduction to homogeneous geodesics, that is, geodesics that are integral curves of Killing vector fields, presenting both classical and modern results, some very recent, many of which are due to the authors. The main focus is on the class of Riemannian manifolds with homogeneous geodesics and on some of its important subclasses. To keep the exposition self-contained the book also includes useful general results not only on geodesic orbit manifolds, but also on smooth and Riemannian manifolds, Lie groups and Lie algebras, homogeneous Riemannian manifolds, and compact homogeneous Riemannian spaces. The intended audience is graduate students and researchers whose work involves differential geometry and transformation groups.

Metric Structures for Riemannian and Non-Riemannian Spaces

Mikhail Gromov
Metric Structures for Riemannian and Non Riemannian Spaces Book PDF

Author : Mikhail Gromov
Publisher : Springer Science & Business Media
Page : 594 pages
File Size : 44,7 Mb
Release : 2007-06-25
Category : Mathematics
ISBN : 9780817645830

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Metric Structures for Riemannian and Non-Riemannian Spaces by Mikhail Gromov Pdf

This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.

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 : 49,6 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.

Riemannian Manifolds of Conullity Two

Eric Boeckx,Lieven Vanhecke,Oldrich Kowalski
Riemannian Manifolds of Conullity Two Book PDF

Author : Eric Boeckx,Lieven Vanhecke,Oldrich Kowalski
Publisher : World Scientific
Page : 320 pages
File Size : 43,5 Mb
Release : 1996-11-09
Category : Mathematics
ISBN : 9789814498555

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Riemannian Manifolds of Conullity Two by Eric Boeckx,Lieven Vanhecke,Oldrich Kowalski Pdf

This book deals with Riemannian manifolds for which the nullity space of the curvature tensor has codimension two. These manifolds are “semi-symmetric spaces foliated by Euclidean leaves of codimension two” in the sense of Z I Szabó. The authors concentrate on the rich geometrical structure and explicit descriptions of these remarkable spaces. Also parallel theories are developed for manifolds of “relative conullity two”. This makes a bridge to a survey on curvature homogeneous spaces introduced by I M Singer. As an application of the main topic, interesting hypersurfaces with type number two in Euclidean space are discovered, namely those which are locally rigid or “almost rigid”. The unifying method is solving explicitly particular systems of nonlinear PDE. Contents:IntroductionDefinition of Semi-Symmetric Spaces and Early DevelopmentLocal Structure of Semi-Symmetric SpacesExplicit Treatment of Foliated Semi-Symmetric SpacesCurvature Homogeneous Semi-Symmetric SpacesAsymptotic Foliations and Algebraic RankThree-Dimensional Riemannian Manifolds of Conullity TwoAsymptotically Foliated Semi-Symmetric SpacesElliptic Semi-Symmetric SpacesComplete Foliated Semi-Symmetric SpacesApplication: Local Rigidity Problems for Hypersurfaces with Type Number Two in IR4Three-Dimensional Riemannian Manifolds of c-Conullity TwoMore about Curvature Homogeneous SpacesBiolographyIndex Readership: Mathematicians and mathematical physicists. keywords:Riemannian Manifold;Curvature Homogeneous Space;Semi-Symmetric Space;Pseudo-Symmetric Space;Asymptotic Foliation;Hypersurface with Type Number Two;Gromov Conjecture;Lichnerowicz Formula;Nomizu Conjecture;Singer Number

The Volume of Vector Fields on Riemannian Manifolds

Olga Gil-Medrano
The Volume of Vector Fields on Riemannian Manifolds Book PDF

Author : Olga Gil-Medrano
Publisher : Springer Nature
Page : 131 pages
File Size : 49,6 Mb
Release : 2023-07-31
Category : Mathematics
ISBN : 9783031368578

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The Volume of Vector Fields on Riemannian Manifolds by Olga Gil-Medrano Pdf

This 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.

Topics in Geometry

Simon Gindikin
Topics in Geometry Book PDF

Author : Simon Gindikin
Publisher : Springer Science & Business Media
Page : 396 pages
File Size : 50,7 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.

Geometric Control Theory and Sub-Riemannian Geometry

Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti
Geometric Control Theory and Sub Riemannian Geometry Book PDF

Author : Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti
Publisher : Springer
Page : 384 pages
File Size : 54,5 Mb
Release : 2014-06-05
Category : Mathematics
ISBN : 9783319021324

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Geometric Control Theory and Sub-Riemannian Geometry by Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti Pdf

Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.

Riemannian Manifolds of Conullity Two

Eric Boeckx,Old?ich Kowalski,Lieven Vanhecke
Riemannian Manifolds of Conullity Two Book PDF

Author : Eric Boeckx,Old?ich Kowalski,Lieven Vanhecke
Publisher : World Scientific
Page : 319 pages
File Size : 54,8 Mb
Release : 1996
Category : Mathematics
ISBN : 9789810227685

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Riemannian Manifolds of Conullity Two by Eric Boeckx,Old?ich Kowalski,Lieven Vanhecke Pdf

This book deals with Riemannian manifolds for which the nullity space of the curvature tensor has codimension two. These manifolds are ?semi-symmetric spaces foliated by Euclidean leaves of codimension two? in the sense of Z I Szab¢. The authors concentrate on the rich geometrical structure and explicit descriptions of these remarkable spaces. Also parallel theories are developed for manifolds of ?relative conullity two?. This makes a bridge to a survey on curvature homogeneous spaces introduced by I M Singer. As an application of the main topic, interesting hypersurfaces with type number two in Euclidean space are discovered, namely those which are locally rigid or ?almost rigid?. The unifying method is solving explicitly particular systems of nonlinear PDE.

Homogeneous Manifolds with Negative Curvature, Part II

Robert Azencott,Edward N. Wilson
Homogeneous Manifolds with Negative Curvature Part II Book PDF

Author : Robert Azencott,Edward N. Wilson
Publisher : American Mathematical Soc.
Page : 102 pages
File Size : 52,8 Mb
Release : 1976
Category : Mathematics
ISBN : 9780821821787

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Homogeneous Manifolds with Negative Curvature, Part II by Robert Azencott,Edward N. Wilson Pdf

This paper is the second in a series dealing with the structure of the full isometry group I(M) for M a connected, simply connected, homogeneous, Riemannian manifold with non-positive sectional curvature. It is shown that every such manifold determines canonically a conjugacy class of subgroups of I(M) which act simply transitively on M. The class of all simply transitive subgroups of I(M) is identified and it is demonstrated that an arbitrary simply transitive subgroup may be modified slightly to produce a subgroup in the canonical class. The class of all connected Lie groups G for which there exists such a manifold M with G isomorphic to the identity connected component of I(M) is identified by means of a list of structural conditions on the Lie algebra of G. Given an arbitrary connected, simply connected Riemannian manifold M together with a given simply transitive group S of isometries, an algorithm is exhibited to explicitly compute the Lie algebra of I(M) from the transported Riemannian data on S.

The Mathematical Heritage Of C F Gauss

George M Rassias
The Mathematical Heritage Of C F Gauss Book PDF

Author : George M Rassias
Publisher : World Scientific
Page : 916 pages
File Size : 54,8 Mb
Release : 1991-09-30
Category : Electronic
ISBN : 9789814603799

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The Mathematical Heritage Of C F Gauss by George M Rassias Pdf

This volume is a collection of original and expository papers in the fields of Mathematics in which Gauss had made many fundamental discoveries. The contributors are all outstanding in their fields and the volume will be of great interest to all research mathematicians, research workers in the history of science, and graduate students in Mathematics and Mathematical Physics.