# Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications

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## Riemannian Submersions Riemannian Maps in Hermitian Geometry and their Applications

Author | : Bayram Sahin |

Publsiher | : Academic Press |

Total Pages | : 360 |

Release | : 2017-01-23 |

ISBN 10 | : 0128044101 |

ISBN 13 | : 9780128044100 |

Language | : EN, FR, DE, ES & NL |

**Riemannian Submersions Riemannian Maps in Hermitian Geometry and their Applications Book Review:**

Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications is a rich and self-contained exposition of recent developments in Riemannian submersions and maps relevant to complex geometry, focusing particularly on novel submersions, Hermitian manifolds, and K\{a}hlerian manifolds. Riemannian submersions have long been an effective tool to obtain new manifolds and compare certain manifolds within differential geometry. For complex cases, only holomorphic submersions function appropriately, as discussed at length in Falcitelli, Ianus and Pastore’s classic 2004 book. In this new book, Bayram Sahin extends the scope of complex cases with wholly new submersion types, including Anti-invariant submersions, Semi-invariant submersions, slant submersions, and Pointwise slant submersions, also extending their use in Riemannian maps. The work obtains new properties of the domain and target manifolds and investigates the harmonicity and geodesicity conditions for such maps. It also relates these maps with discoveries in pseudo-harmonic maps. Results included in this volume should stimulate future research on Riemannian submersions and Riemannian maps. Systematically reviews and references modern literature in Riemannian maps Provides rigorous mathematical theory with applications Presented in an accessible reading style with motivating examples that help the reader rapidly progress

## Manifolds II

Author | : Paul Bracken |

Publsiher | : BoD – Books on Demand |

Total Pages | : 146 |

Release | : 2019-05-22 |

ISBN 10 | : 1838803092 |

ISBN 13 | : 9781838803094 |

Language | : EN, FR, DE, ES & NL |

**Manifolds II Book Review:**

Differential geometry is a very active field of research and has many applications to areas such as physics, in particular gravity. The chapters in this book cover a number of subjects that will be of interest to workers in these areas. It is hoped that these chapters will be able to provide a useful resource for researchers with regard to current fields of research in this important area.

## Differential Geometry of Lightlike Submanifolds

Author | : Krishan Duggal,Bayram Sahin |

Publsiher | : Birkhäuser |

Total Pages | : 488 |

Release | : 2011-04-08 |

ISBN 10 | : 9783034602525 |

ISBN 13 | : 3034602529 |

Language | : EN, FR, DE, ES & NL |

**Differential Geometry of Lightlike Submanifolds Book Review:**

This book presents research on the latest developments in differential geometry of lightlike (degenerate) subspaces. The main focus is on hypersurfaces and a variety of submanifolds of indefinite Kählerian, Sasakian and quaternion Kähler manifolds.

## Differential Geometry of Lightlike Submanifolds

Author | : Krishan L. Duggal,Bayram Sahin |

Publsiher | : Springer Science & Business Media |

Total Pages | : 488 |

Release | : 2011-02-02 |

ISBN 10 | : 3034602510 |

ISBN 13 | : 9783034602518 |

Language | : EN, FR, DE, ES & NL |

**Differential Geometry of Lightlike Submanifolds Book Review:**

This book presents research on the latest developments in differential geometry of lightlike (degenerate) subspaces. The main focus is on hypersurfaces and a variety of submanifolds of indefinite Kählerian, Sasakian and quaternion Kähler manifolds.

## Pseudo Riemannian Geometry Invariants and Applications

Author | : Bang-Yen Chen |

Publsiher | : World Scientific |

Total Pages | : 512 |

Release | : 2011-03-23 |

ISBN 10 | : 9814462489 |

ISBN 13 | : 9789814462488 |

Language | : EN, FR, DE, ES & NL |

**Pseudo Riemannian Geometry Invariants and Applications Book Review:**

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;Î´-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

## Riemannian Submersions and Related Topics

Author | : Maria Falcitelli,Anna Maria Pastore,Stere Ianus? |

Publsiher | : World Scientific |

Total Pages | : 277 |

Release | : 2004 |

ISBN 10 | : 9812562338 |

ISBN 13 | : 9789812562333 |

Language | : EN, FR, DE, ES & NL |

**Riemannian Submersions and Related Topics Book Review:**

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.

## Spectral Geometry Riemannian Submersions and the Gromov Lawson Conjecture

Author | : Peter B. Gilkey,John V Leahy,JeongHyeong Park |

Publsiher | : CRC Press |

Total Pages | : 296 |

Release | : 1999-07-27 |

ISBN 10 | : 9780849382772 |

ISBN 13 | : 0849382777 |

Language | : EN, FR, DE, ES & NL |

**Spectral Geometry Riemannian Submersions and the Gromov Lawson Conjecture Book Review:**

This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change. Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area. Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture. Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.

## Harmonic Morphisms Between Riemannian Manifolds

Author | : Paul Baird,Professor of Mathematics Paul Baird,John C. Wood,John C.. Wood,Professor of Pure Mathematics John C Wood |

Publsiher | : Oxford University Press |

Total Pages | : 520 |

Release | : 2003 |

ISBN 10 | : 9780198503620 |

ISBN 13 | : 0198503628 |

Language | : EN, FR, DE, ES & NL |

**Harmonic Morphisms Between Riemannian Manifolds Book Review:**

This is the first account in book form of the theory of harmonic morphisms between Riemannian manifolds. Harmonic morphisms are maps which preserve Laplace's equation. They can be characterized as harmonic maps which satisfy an additional first order condition. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a Riemann surface. There are connections with many concepts in differential geometry, for example, Killing fields, geodesics, foliations, Clifford systems, twistor spaces, Hermitian structures, iso-parametric mappings, and Einstein metrics and also the Brownain pathpreserving maps of probability theory. Giving a complete account of the fundamental aspects of the subject, this book is self-contained, assuming only a basic knowledge of differential geometry.

## Encyclopaedia of Mathematics

Author | : Michiel Hazewinkel |

Publsiher | : Springer Science & Business Media |

Total Pages | : 632 |

Release | : 2012-12-06 |

ISBN 10 | : 9401512795 |

ISBN 13 | : 9789401512794 |

Language | : EN, FR, DE, ES & NL |

**Encyclopaedia of Mathematics Book Review:**

This is the second supplementary volume to Kluwer's highly acclaimed eleven-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing eleven volumes, and together these twelve volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.

## Geometry of Cauchy Riemann Submanifolds

Author | : Sorin Dragomir,Mohammad Hasan Shahid,Falleh R. Al-Solamy |

Publsiher | : Springer |

Total Pages | : 390 |

Release | : 2016-05-31 |

ISBN 10 | : 9811009163 |

ISBN 13 | : 9789811009167 |

Language | : EN, FR, DE, ES & NL |

**Geometry of Cauchy Riemann Submanifolds Book Review:**

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.

## Two Reports on Harmonic Maps

Author | : James Eells,Luc Lemaire |

Publsiher | : World Scientific |

Total Pages | : 228 |

Release | : 1995-03-29 |

ISBN 10 | : 9814502928 |

ISBN 13 | : 9789814502924 |

Language | : EN, FR, DE, ES & NL |

**Two Reports on Harmonic Maps Book Review:**

Harmonic maps between Riemannian manifolds are solutions of systems of nonlinear partial differential equations which appear in different contexts of differential geometry. They include holomorphic maps, minimal surfaces, σ-models in physics. Recently, they have become powerful tools in the study of global properties of Riemannian and Kählerian manifolds. A standard reference for this subject is a pair of Reports, published in 1978 and 1988 by James Eells and Luc Lemaire. This book presents these two reports in a single volume with a brief supplement reporting on some recent developments in the theory. It is both an introduction to the subject and a unique source of references, providing an organized exposition of results spread throughout more than 800 papers. Contents:IntroductionOperations on Vector BundlesHarmonic MapsComposition PropertiesMaps into Manifolds of Nonpositive (≤ 0) CurvatureThe Existence Theorem for Riem N ≤ 0Maps into Flat ManifoldsHarmonic Maps between SpheresHolomorphic MapsHarmonic Maps of a SurfaceHarmonic Maps between SurfacesHarmonic Maps of Manifolds with Boundary Readership: Mathematicians and mathematical physicists. keywords:Harmonic Maps;Minimal Immersions;Totally Geodesic Maps;Kaehler Manifold;(1,1)-Geodesic Map;Dilatation;Nonpositive Sectional Curvature;Holomorphic Map;Teichmueller Map;Twistor Construction “… an interesting account of the progress made in the theory of harmonic maps until the year 1988 … this master-piece work will serve as an influence and good reference in the very active subject of harmonic maps both from the points of view of theory and applications.” Mathematics Abstracts

## Mathematical Reviews

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 2003 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : UVA:X006180631 |

Language | : EN, FR, DE, ES & NL |

**Mathematical Reviews Book Review:**

## Differential Geometry

Author | : Vagn Lundsgaard Hansen |

Publsiher | : Springer |

Total Pages | : 288 |

Release | : 2006-11-15 |

ISBN 10 | : 3540472495 |

ISBN 13 | : 9783540472490 |

Language | : EN, FR, DE, ES & NL |

**Differential Geometry Book Review:**

The Nordic Summer School 1985 presented to young researchers the mathematical aspects of the ongoing research stemming from the study of field theories in physics and the differential geometry of fibre bundles in mathematics. The volume includes papers, often with original lines of attack, on twistor methods for harmonic maps, the differential geometric aspects of Yang-Mills theory, complex differential geometry, metric differential geometry and partial differential equations in differential geometry. Most of the papers are of lasting value and provide a good introduction to their subject.

## Structures on Manifolds

Author | : K Yano,M Kon |

Publsiher | : World Scientific |

Total Pages | : 520 |

Release | : 1985-02-01 |

ISBN 10 | : 9814602809 |

ISBN 13 | : 9789814602808 |

Language | : EN, FR, DE, ES & NL |

**Structures on Manifolds Book Review:**

Contents: Riemannian ManifoldsSubmanifolds of Riemannian ManifoldsComplex ManifoldsSubmanifolds of Kaehlerian ManifoldsContact ManifoldsSubmanifolds of Sasakian Manifoldsf-StructuresProduct ManifoldsSubmersions Readership: Mathematicians. Keywords:Riemannian Manifold;Submanifold;Complex Manifold;Contact Manifold;Kaehlerian Manifold;Sasakian Manifold;Anti-Invariant Submanifold;CR Submanifold;Contact CR Submanifold;Submersion

## Differential Geometry and Differential Equations

Author | : Chaohao Gu,Marcel Berger,Robert L. Bryant |

Publsiher | : Springer |

Total Pages | : 246 |

Release | : 2006-11-15 |

ISBN 10 | : 3540478833 |

ISBN 13 | : 9783540478836 |

Language | : EN, FR, DE, ES & NL |

**Differential Geometry and Differential Equations Book Review:**

The DD6 Symposium was, like its predecessors DD1 to DD5 both a research symposium and a summer seminar and concentrated on differential geometry. This volume contains a selection of the invited papers and some additional contributions. They cover recent advances and principal trends in current research in differential geometry.

## Revue Roumaine de Math matiques Pures Et Appliqu es

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 2007 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : UOM:39015072688552 |

Language | : EN, FR, DE, ES & NL |

**Revue Roumaine de Math matiques Pures Et Appliqu es Book Review:**

## A Panoramic View of Riemannian Geometry

Author | : Marcel Berger |

Publsiher | : Springer Science & Business Media |

Total Pages | : 824 |

Release | : 2012-12-06 |

ISBN 10 | : 3642182453 |

ISBN 13 | : 9783642182457 |

Language | : EN, FR, DE, ES & NL |

**A Panoramic View of Riemannian Geometry Book Review:**

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

## An Introduction to Riemannian Geometry

Author | : Leonor Godinho,José Natário |

Publsiher | : Springer |

Total Pages | : 467 |

Release | : 2014-07-26 |

ISBN 10 | : 3319086669 |

ISBN 13 | : 9783319086668 |

Language | : EN, FR, DE, ES & NL |

**An Introduction to Riemannian Geometry Book Review:**

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

## Lightlike Submanifolds of Semi Riemannian Manifolds and Applications

Author | : Krishan L. Duggal,Aurel Bejancu |

Publsiher | : Springer Science & Business Media |

Total Pages | : 303 |

Release | : 2013-04-17 |

ISBN 10 | : 9401720894 |

ISBN 13 | : 9789401720892 |

Language | : EN, FR, DE, ES & NL |

**Lightlike Submanifolds of Semi Riemannian Manifolds and Applications Book Review:**

This book is about the light like (degenerate) geometry of submanifolds needed to fill a gap in the general theory of submanifolds. The growing importance of light like hypersurfaces in mathematical physics, in particular their extensive use in relativity, and very limited information available on the general theory of lightlike submanifolds, motivated the present authors, in 1990, to do collaborative research on the subject matter of this book. Based on a series of author's papers (Bejancu [3], Bejancu-Duggal [1,3], Dug gal [13], Duggal-Bejancu [1,2,3]) and several other researchers, this volume was conceived and developed during the Fall '91 and Fall '94 visits of Bejancu to the University of Windsor, Canada. The primary difference between the lightlike submanifold and that of its non degenerate counterpart arises due to the fact that in the first case, the normal vector bundle intersects with the tangent bundle of the submanifold. Thus, one fails to use, in the usual way, the theory of non-degenerate submanifolds (cf. Chen [1]) to define the induced geometric objects (such as linear connection, second fundamental form, Gauss and Weingarten equations) on the light like submanifold. Some work is known on null hypersurfaces and degenerate submanifolds (see an up-to-date list of references on pages 138 and 140 respectively). Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of an up-to-date information on null curves, lightlike hypersur faces and submanifolds, consistent with the theory of non-degenerate submanifolds.

## Geometry of Submanifolds

Author | : Bang-Yen Chen |

Publsiher | : Courier Dover Publications |

Total Pages | : 192 |

Release | : 2019-06-12 |

ISBN 10 | : 0486832783 |

ISBN 13 | : 9780486832784 |

Language | : EN, FR, DE, ES & NL |

**Geometry of Submanifolds Book Review:**

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.