Pseudo Riemannian Geometry δ Invariants And Applications

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Pseudo-Riemannian Geometry, [delta]-invariants and Applications

Bang-yen Chen
Pseudo Riemannian Geometry delta invariants and Applications Book PDF

Author : Bang-yen Chen
Publisher : World Scientific
Page : 510 pages
File Size : 40,9 Mb
Release : 2011
Category : Mathematics
ISBN : 9789814329637

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Pseudo-Riemannian Geometry, [delta]-invariants and Applications by Bang-yen Chen Pdf

The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included.The second part of this book is on ë-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as ë-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between ë-invariants and the main extrinsic invariants. Since then many new results concerning these ë-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.

Pseudo-Riemannian Geometry, δ-Invariants and Applications

Bang-Yen Chen
Pseudo Riemannian Geometry Invariants and Applications Book PDF

Author : Bang-Yen Chen
Publisher : World Scientific
Page : 512 pages
File Size : 45,9 Mb
Release : 2011-03-23
Category : Mathematics
ISBN : 9789814462488

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Pseudo-Riemannian Geometry, δ-Invariants and Applications by Bang-Yen Chen Pdf

The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on δ-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades. Contents:Pseudo-Riemannian ManifoldsBasics on Pseudo-Riemannian SubmanifoldsSpecial Pseudo-Riemannian SubmanifoldsWarped Products and Twisted ProductsRobertson–Walker SpacetimesHodge Theory, Elliptic Differential Operators and Jacobi's Elliptic FunctionsSubmanifolds of Finite TypeTotal Mean CurvaturePseudo-Kähler ManifoldsPara-Kähler ManifoldsPseudo-Riemannian SubmersionsContact Metric Manifolds and Submanifoldsδ-Invariants, Inequalities and Ideal ImmersionsSome Applications of δ-InvariantsApplications to Kähler and Para-Kähler GeometryApplications to Contact GeometryApplications to Affine GeometryApplications to Riemannian SubmersionsNearly Kähler Manifolds and Nearly Kähler S6(1)δ(2)-Ideal Immersions Readership: Graduate and PhD students in differential geometry and related fields; researchers in differential geometry and related fields; theoretical physicists. Keywords:Pseudo-Riemannian Submanifold;δ-Invariants;Spacetimes;Submersion;Lagrangian Submanifolds;Sasakian Manifold;Total Mean Curvature;Submanifold of Finite Type;Affine HypersurfaceKey Features:This is the only book that provides general results on pseudo-Riemannian submanifoldsThis is the only book that provides detailed account on δ-invariantsAt the beginning of each chapter, historical background is providedReviews: “This book gives an extensive and in-depth overview of the theory of pseudo-Riemannian submanifolds and of the delta-invariants. It is written in an accessible and quite self-contained way. Hence it is recommendable for a very broad audience of students and mathematicians interested in the geometry of submanifolds.” Mathematical Reviews “This books is an extensive and comprehensive survey on pseudo–Riemannian submanifolds and δ–invariants as well as their applications. In every aspect, this is an excellent book, invaluable both for learning the topic and a reference. Therefore, it should be strongly recommended for students and mathematicians interested in the geometry of pseudo-Riemannian submanifolds.” Zentralblatt MATH

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 : 40,8 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.

Geometry of Cauchy-Riemann Submanifolds

Sorin Dragomir,Mohammad Hasan Shahid,Falleh R. Al-Solamy
Geometry of Cauchy Riemann Submanifolds Book PDF

Author : Sorin Dragomir,Mohammad Hasan Shahid,Falleh R. Al-Solamy
Publisher : Springer
Page : 390 pages
File Size : 53,6 Mb
Release : 2016-05-31
Category : Mathematics
ISBN : 9789811009167

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Geometry of Cauchy-Riemann Submanifolds by Sorin Dragomir,Mohammad Hasan Shahid,Falleh R. Al-Solamy Pdf

This book gathers contributions by respected experts on the theory of isometric immersions between Riemannian manifolds, and focuses on the geometry of CR structures on submanifolds in Hermitian manifolds. CR structures are a bundle theoretic recast of the tangential Cauchy–Riemann equations in complex analysis involving several complex variables. The book covers a wide range of topics such as Sasakian geometry, Kaehler and locally conformal Kaehler geometry, the tangential CR equations, Lorentzian geometry, holomorphic statistical manifolds, and paraquaternionic CR submanifolds. Intended as a tribute to Professor Aurel Bejancu, who discovered the notion of a CR submanifold of a Hermitian manifold in 1978, the book provides an up-to-date overview of several topics in the geometry of CR submanifolds. Presenting detailed information on the most recent advances in the area, it represents a useful resource for mathematicians and physicists alike.

Differential Geometry Of Warped Product Manifolds And Submanifolds

Chen Bang-yen
Differential Geometry Of Warped Product Manifolds And Submanifolds Book PDF

Author : Chen Bang-yen
Publisher : World Scientific
Page : 516 pages
File Size : 50,5 Mb
Release : 2017-05-29
Category : Mathematics
ISBN : 9789813208940

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Differential Geometry Of Warped Product Manifolds And Submanifolds by Chen Bang-yen Pdf

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson–Walker models, are warped product manifolds. The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson–Walker's and Schwarzschild's. The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century. The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.

Differential Geometry

Ion Mihai
Differential Geometry Book PDF

Author : Ion Mihai
Publisher : MDPI
Page : 166 pages
File Size : 49,8 Mb
Release : 2019-11-21
Category : Mathematics
ISBN : 9783039218004

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Differential Geometry by Ion Mihai Pdf

The present book contains 14 papers published in the Special Issue “Differential Geometry” of the journal Mathematics. They represent a selection of the 30 submissions. This book covers a variety of both classical and modern topics in differential geometry. We mention properties of both rectifying and affine curves, the geometry of hypersurfaces, angles in Minkowski planes, Euclidean submanifolds, differential operators and harmonic forms on Riemannian manifolds, complex manifolds, contact manifolds (in particular, Sasakian and trans-Sasakian manifolds), curvature invariants, and statistical manifolds and their submanifolds (in particular, Hessian manifolds). We wish to mention that among the authors, there are both well-known geometers and young researchers. The authors are from countries with a tradition in differential geometry: Belgium, China, Greece, Japan, Korea, Poland, Romania, Spain, Turkey, and United States of America. Many of these papers were already cited by other researchers in their articles. This book is useful for specialists in differential geometry, operator theory, physics, and information geometry as well as graduate students in mathematics.

Recent Advances in the Geometry of Submanifolds

Bogdan D. Suceavă,Alfonso Carriazo,Yun Myung Oh,Joeri Van der Veken
Recent Advances in the Geometry of Submanifolds Book PDF

Author : Bogdan D. Suceavă,Alfonso Carriazo,Yun Myung Oh,Joeri Van der Veken
Publisher : American Mathematical Soc.
Page : 209 pages
File Size : 51,9 Mb
Release : 2016-09-14
Category : Differential geometry -- Local differential geometry -- Local submanifolds
ISBN : 9781470422981

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Recent Advances in the Geometry of Submanifolds by Bogdan D. Suceavă,Alfonso Carriazo,Yun Myung Oh,Joeri Van der Veken Pdf

This volume contains the proceedings of the AMS Special Session on Geometry of Submanifolds, held from October 25–26, 2014, at San Francisco State University, San Francisco, CA, and the AMS Special Session on Recent Advances in the Geometry of Submanifolds: Dedicated to the Memory of Franki Dillen (1963–2013), held from March 14–15, 2015, at Michigan State University, East Lansing, Ml. The focus of the volume is on recent studies of submanifolds of Riemannian, semi-Riemannian, Kaehlerian and contact manifolds. Some of these use techniques in classical differential geometry, while others use methods from ordinary differential equations, geometric analysis, or geometric PDEs. By brainstorming on the fundamental problems and exploring a large variety of questions studied in submanifold geometry, the editors hope to provide mathematicians with a working tool, not just a collection of individual contributions. This volume is dedicated to the memory of Franki Dillen, whose work in submanifold theory attracted the attention of and inspired many geometers.

Geometry of Submanifolds and Homogeneous Spaces

Andreas Arvanitoyeorgos,George Kaimakamis
Geometry of Submanifolds and Homogeneous Spaces Book PDF

Author : Andreas Arvanitoyeorgos,George Kaimakamis
Publisher : MDPI
Page : 128 pages
File Size : 55,5 Mb
Release : 2020-01-03
Category : Mathematics
ISBN : 9783039280001

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Geometry of Submanifolds and Homogeneous Spaces by Andreas Arvanitoyeorgos,George Kaimakamis Pdf

The present Special Issue of Symmetry is devoted to two important areas of global Riemannian geometry, namely submanifold theory and the geometry of Lie groups and homogeneous spaces. Submanifold theory originated from the classical geometry of curves and surfaces. Homogeneous spaces are manifolds that admit a transitive Lie group action, historically related to F. Klein's Erlangen Program and S. Lie's idea to use continuous symmetries in studying differential equations. In this Special Issue, we provide a collection of papers that not only reflect some of the latest advancements in both areas, but also highlight relations between them and the use of common techniques. Applications to other areas of mathematics are also considered.

Geometry of Submanifolds

Bang-Yen Chen
Geometry of Submanifolds Book PDF

Author : Bang-Yen Chen
Publisher : Courier Dover Publications
Page : 193 pages
File Size : 51,6 Mb
Release : 2019-06-12
Category : Mathematics
ISBN : 9780486832784

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Geometry of Submanifolds by Bang-Yen Chen Pdf

The first two chapters of this frequently cited reference provide background material in Riemannian geometry and the theory of submanifolds. Subsequent chapters explore minimal submanifolds, submanifolds with parallel mean curvature vector, conformally flat manifolds, and umbilical manifolds. The final chapter discusses geometric inequalities of submanifolds, results in Morse theory and their applications, and total mean curvature of a submanifold. Suitable for graduate students and mathematicians in the area of classical and modern differential geometries, the treatment is largely self-contained. Problems sets conclude each chapter, and an extensive bibliography provides background for students wishing to conduct further research in this area. This new edition includes the author's corrections.

Global Riemannian Geometry: Curvature and Topology

Ana Hurtado,Steen Markvorsen,Maung Min-Oo,Vicente Palmer
Global Riemannian Geometry Curvature and Topology Book PDF

Author : Ana Hurtado,Steen Markvorsen,Maung Min-Oo,Vicente Palmer
Publisher : Springer Nature
Page : 121 pages
File Size : 52,9 Mb
Release : 2020-08-19
Category : Mathematics
ISBN : 9783030552930

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Global Riemannian Geometry: Curvature and Topology by Ana Hurtado,Steen Markvorsen,Maung Min-Oo,Vicente Palmer Pdf

This book contains a clear exposition of two contemporary topics in modern differential geometry: distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature the application of the Lichnerowicz formula for Dirac operators to the study of Gromov's invariants to measure the K-theoretic size of a Riemannian manifold. It is intended for both graduate students and researchers.

Riemannian Geometry

Sylvestre Gallot,Dominique Hulin,Jacques Lafontaine
Riemannian Geometry Book PDF

Author : Sylvestre Gallot,Dominique Hulin,Jacques Lafontaine
Publisher : Springer Science & Business Media
Page : 346 pages
File Size : 42,8 Mb
Release : 2004-07-30
Category : Mathematics
ISBN : 3540204938

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Riemannian Geometry by Sylvestre Gallot,Dominique Hulin,Jacques Lafontaine Pdf

This 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.

Aspects of Differential Geometry I

Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Aspects of Differential Geometry I Book PDF

Author : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 140 pages
File Size : 52,6 Mb
Release : 2022-05-31
Category : Mathematics
ISBN : 9783031024078

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Aspects of Differential Geometry I by Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo Pdf

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new. Table of Contents: Preface / Acknowledgments / Basic Notions and Concepts / Manifolds / Riemannian and Pseudo-Riemannian Geometry / Bibliography / Authors' Biographies / Index

Singular Semi-Riemannian Geometry

D.N. Kupeli
Singular Semi Riemannian Geometry Book PDF

Author : D.N. Kupeli
Publisher : Springer Science & Business Media
Page : 181 pages
File Size : 53,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401587617

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Singular Semi-Riemannian Geometry by D.N. Kupeli Pdf

This book is an exposition of "Singular Semi-Riemannian Geometry"- the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. The main topic of interest is those cases where the metric tensor is assumed to be nondegenerate. In the literature, manifolds with degenerate metric tensors have been studied extrinsically as degenerate submanifolds of semi Riemannian manifolds. One major aspect of this book is first to study the intrinsic structure of a manifold with a degenerate metric tensor and then to study it extrinsically by considering it as a degenerate submanifold of a semi-Riemannian manifold. This book is divided into three parts. Part I deals with singular semi Riemannian manifolds in four chapters. In Chapter I, the linear algebra of indefinite real inner product spaces is reviewed. In general, properties of certain geometric tensor fields are obtained purely from the algebraic point of view without referring to their geometric origin. Chapter II is devoted to a review of covariant derivative operators in real vector bundles. Chapter III is the main part of this book where, intrinsically, the Koszul connection is introduced and its curvature identities are obtained. In Chapter IV, an application of Chapter III is made to degenerate submanifolds of semi-Riemannian manifolds and Gauss, Codazzi and Ricci equations are obtained. Part II deals with singular Kahler manifolds in four chapters parallel to Part I.

Riemannian Submersions and Related Topics

Maria Falcitelli,Anna Maria Pastore,Stere Ianus?
Riemannian Submersions and Related Topics Book PDF

Author : Maria Falcitelli,Anna Maria Pastore,Stere Ianus?
Publisher : World Scientific
Page : 292 pages
File Size : 49,7 Mb
Release : 2004
Category : Mathematics
ISBN : 9789812562333

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Riemannian Submersions and Related Topics by Maria Falcitelli,Anna Maria Pastore,Stere Ianus? Pdf

This book provides the first-ever systematic introduction to thetheory of Riemannian submersions, which was initiated by BarrettO''Neill and Alfred Gray less than four decades ago. The authorsfocus their attention on classification theorems when the total spaceand the fibres have nice geometric properties.

Aspects of Differential Geometry II

Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Aspects of Differential Geometry II Book PDF

Author : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 143 pages
File Size : 46,8 Mb
Release : 2022-06-01
Category : Mathematics
ISBN : 9783031024085

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Aspects of Differential Geometry II by Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo Pdf

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.