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

By Bayram Sahin

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

• Language: English
• Publisher: Academic Press
• Release date: Jan 23, 2017
• ISBN: 9780128044100

Bayram Sahin

Bayram Sahin was born in Malatya, Turkey. He received his Bachelor’s degree from Ege University, Izmir, Turkey, in 1993 and his Ph.D from Inonu University in 2000. After graduating, Dr. Sahin worked as a post-doctoral fellow at University of Windsor from March 2003 to September 2003 and a research scholar from June 2007 to September 2007. He has led several TUBITAK funded projects at the interface of manifold theory and maps and he has written (or co-authored) eighty-one academic papers. He is the author of the monograph “Differential Geometry of Lightlike Submanifolds” (2010) and the editors of Turkish Journal of Mathematics and Mediterranean Journal of Mathematics. He is the recipient of Masatoshi Gunduz Ikeda research award at 2006. He is now a professor of Mathematics at Ege University, Turkey.

Book preview

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

Bayram Şahin

Department of Mathematics, Ege University, Izmir, Turkey

Table of Contents

Cover image

Title page

Copyright

Dedication

Acknowledgments

Preface

About the Author

Chapter 1: Basic Geometric Structures on Manifolds

Abstract

1 Riemannian manifolds and related topics

2 Vector bundles

3 Riemannian submanifolds and distributions

4 Riemannian submersions

5 Certain product structures on manifolds

6 Geometric structures along a map

Chapter 2: Applications of Riemannian Submersions

Abstract

1 Applications of Riemannian submersions in robotic theory

2 Kaluza-Klein theory

Chapter 3: Riemannian submersions From Almost Hermitian Manifolds

Abstract

1 Almost Hermitian manifolds

2 Holomorphic submersions and invariant submersions

3 Anti-invariant Riemannian submersions

4 Semi-invariant submersions

5 Generic Riemannian submersions

6 Slant submersions

7 Semi-slant submersions

8 Hemi-slant submersions

9 Pointwise slant submersions

10 Einstein metrics on the total space of an anti-invariant submersion

11 Clairaut submersions from almost Hermitian manifolds

Chapter 4: Riemannian Maps

Abstract

1 Riemannian maps

2 Geometric structures along Riemannian maps

3 Totally geodesic Riemannian maps

4 Umbilical Riemannian maps

5 Harmonicity of Riemannian maps

6 Clairaut Riemannian maps

7 Circles along Riemannian maps

8 Chen first inequality for Riemannian maps

9 Einstein metrics on the total space of a Riemannian map

Chapter 5: Riemannian Maps From Almost Hermitian Manifolds

Abstract

1 Holomorphic Riemannian maps from almost Hermitian manifolds

2 Anti-invariant Riemannian maps from almost Hermitian manifolds

3 Semi-invariant Riemannian maps from almost Hermitian manifolds

4 Generic Riemannian maps from almost Hermitian manifolds

5 Slant Riemannian maps from almost Hermitian manifolds

6 Semi-slant Riemannian maps from almost Hermitian manifolds

7 Hemi-slant Riemannian maps from almost Hermitian manifolds

Chapter 6: Riemannian Maps To Almost Hermitian Manifolds

Abstract

1 Invariant Riemannian maps to almost Hermitian manifolds

2 Anti-invariant Riemannian maps to almost Hermitian manifolds

3 Semi-invariant Riemannian maps to almost Hermitian manifolds

4 Generic Riemannian maps to Kähler manifolds

5 Slant Riemannian maps to Kähler manifolds

6 Semi-slant Riemannian maps to Kähler manifolds

7 Hemi-slant Riemannian maps to Kähler manifolds

Bibliography

Index

Copyright

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Dedication

To Fulya, Cemre, and İdil

Acknowledgments

Bayram Şahin August 2016

First of all, I would like to thank my precious wife Fulya for her love, understanding, and support throughout.

I would like to express my deep gratitude to my sister Sebahat and my elder brother Emin for their patience and loyalty.

I gratefully acknowledge my appreciation and gratitude to various colleagues who have collaborated with me on the research relevant to this book. I also gratefully acknowledge the research funding I have received over the years from The Scientific and Technological Research Council of Turkey (TÜBİTAK).

I am thankful to all authors of books and articles whose work I have used in preparing this book.

Last, but not least, I am grateful to the publishers for their effective cooperation and their excellent care in publishing this volume. I thank Susan Ikeda and Graham Nisbet at Elsevier for guidance throughout the publishing process. Any further comments and suggestions by the readers will be gratefully received.

Preface

Bayram Şahin, Ege University August 2016

Smooth maps between Riemannian manifolds are useful for comparing geometric structures between two manifolds. As is indicated in [122], a major flaw in Riemannian geometry (compared to other subjects) is a shortage of suitable types of maps between Riemannian manifolds that will compare their geometric properties. In Riemannian geometry, there are two basic maps; isometric immersions and Riemannian submersions. Isometric immersions (Riemannian submanifolds) are basic such maps between Riemannian manifolds and they are characterized by their Riemannian metrics and Jacobian matrices. More precisely, a smooth map F : (M, gM) → (N, gN) between Riemannian manifolds (M, gM) and (N, gN) is called an isometric immersion (submanifold) if F* is injective and

for vector fields X, Y tangent to M; here F* denotes the derivative map. A smooth map F : (M1, g1) → (M2, g2) is called a Riemannian submersion if F* is onto and it satisfies the above equation for vector fields tangent to the horizontal space (kerF*)⊥. Riemannian submersions between Riemannian manifolds were first studied by O’Neill [205] and Gray [125]; see also [31], [80], and [111]. The differential geometry of isometric immersions is well known and available in many textbooks, (see, for example [64], [69], [70], [74], [87] and [310]). However, the theory of isometric immersions (or submanifold theory) is still a very active research field. On the other hand, compared with isometric immersions, the geometry of Riemannian submersions has not been studied extensively. This theory is yet available in a few textbooks very recently (see [80] and [111]).

One of the most active and important branches of differential geometry of submanifold theory is the theory of submanifolds of Kähler manifolds. In the early 1930s, Riemannian geometry and the theory of complex variables were synthesized by Kähler [154], which developed (during 1950) into the complex manifold theory [104,190]. A Riemann surface, Cn and its projective space CPn − 1 are simple examples of the complex manifolds. This interrelation between the above two main branches of mathematics developed into what is now known as Kählerian geometry. Almost complex manifolds [306] and their complex, totally real, CR, generic, slant, semi-slant, and hemi-slant submanifolds [26, 53, 68, 70, 83, 202, 214, 238, 308] are some of the most interesting topics of Riemannian geometry. It is known that the complex techniques in relativity have been also very effective tools for understanding spacetime geometry [174]. Indeed, complex manifolds have two interesting classes of Kähler manifolds. One is Calabi-Yau manifolds, which have their applications in superstring theory [51]. The other one is Teichmüller spaces applicable to relativity [288]. It is also important to note that CR-structures have been extensively used in spacetime geometry of relativity [223]. For complex methods in general relativity, see [108].

such that the almost complex structure J carries the tangent space of the submanifold at each point into its normal space and the main properties of such submanifolds established in [83], [202], and [308]. On the other hand, CR-submanifolds were defined by Bejancu [26] as a generalization of complex and totally real submanifolds. By a CR submanifold we mean a real submanifold M , J, ), carrying a J-invariant distribution D (i.e., JD = D-orthogonal complement is J-anti-invariant (i.e., JD⊥ ⊆ T(M)⊥), where T (M)⊥ → M is the normal bundle of M . The CR submanifolds were introduced as an umbrella of a variety (such as invariant, anti-invariant and anti-holomorphic) of submanifolds. A CR-submanifold is called proper if it is neither a complex nor a totally real submanifold. The geometry of CR-submanifolds has been studied in several papers since 1978. Later generic submanifolds were introduced by Chen by releasing an anti-invariant condition on the distribution D⊥. As another generalization of holomorphic submanifolds and totally real submanifolds, slant submanifolds were introduced and studied by Chen in [70]. Recently, semi-slant submanifolds ([214]) and hemi-slant submanifolds ([53], [238]) have been also introduced. Details on these may be seen in [26, 69, 74, 310, 311].

Riemannian submersions as a dual notion of isometric immersions have been studied in complex settings in the early 1970s. First note that a smooth map ϕ : M → N between almost complex manifolds (M, J) and (N, for X ∈ Γ(TM), where J are complex structures of M and N, respectively. As an analogue of holomorphic submanifolds and almost complex maps, Watson [300] defined almost Hermitian submersions between almost Hermitian manifolds and he showed that the base manifold and each fiber have the same kind of structure as the total space, in most cases. We note that almost Hermitian submersions have been extended to the almost contact manifolds [84], locally conformal Kähler manifolds [183], quaternion Kähler manifolds [146], paraquaternionic manifolds [50], statistical manifolds [293], almost product manifolds [131], and paracontact manifolds [137]; see [111] for holomorphic submersions on complex manifolds and their extensions to other manifolds. We also note that Riemannian submersions have their applications in spacetime of unified theory and robotics. In the theory of the Klauza-Klein type, a general solution of the non-linear sigma model is given by Riemannian submersions from the extra-dimensional space to the space in which the scalar fields of the nonlinear sigma model take values; for details, see [111]. In robotic theory, the forward kinematic map can be given by Riemannian submersions; see [12] for details. Moreover, modern Kaluza-Klein theories are given in terms of principal fiber bundle and Riemannian submersion [303].

Although holomorphic submersions, as a corresponding version of holomorphic submanifolds, have been studied by many authors, the other versions of submanifolds have not been studied in the submersion theory. Therefore we introduced and studied anti-invariant Riemannian submersions in [239]. It is shown that holomorphic submersions are useful to determine the geometric structure of the base space; however, anti-invariant submersions are useful to study the geometry of the total space. As a generalization of holomorphic submersions and anti-invariant Riemannian submersions, we introduced semi-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds in [249], then we studied the geometry of such maps. We recall that a Riemannian submersion F from an almost Hermitian manifold (M, JM, gM) with an almost complex structure JM to a Riemannian manifold (N, gN) is called a semi-invariant submersion if the fibers have differentiable distributions D and D⊥ such that D is invariant with respect to JM and its orthogonal complement D. Obviously, invariant Riemannian submersions and anti-invariant Riemannian submersions are semi-invariant submersions with D⊥ = {0} and D = {0}, respectively. As another generalization of invariant submersions and anti-invariant Riemannian submersions, slant submersions were introduced in [245]. Moreover, new submersions and their extensions in this new directions have been studied by various authors; see [3], [5], [6], [7], [8], [30], [105], [106], [131], [132], [133], [134], [135], [136], [169], [171], [172], [215], [216], [219], [221], [248], [265], [283], [284] and [286].

Both these branches, Riemannian submanifolds and Riemannian submersions, are extremely important in differential geometry and as well as in applications in mechanics and general theory of relativity. One of the interesting approach is to study the most general case, which includes both the geometry of submanifolds as well as Riemannian submersions, and such a study was initiated by Fischer [117], where he defines Riemannian maps between Riemannian manifolds as a generalization of the notions of isometric immersions and Riemannian submersions as follows. Let F : (M, gM) → (N, gN) be a smooth map between Riemannian manifolds. Then we denote the kernel space of F* at p ∈ M by kerF*p to kerF*p. Then the tangent space of M at p ∈ M has the following decomposition:

We denote the range of F* at p ∈ M by rangeF*p and consider the orthogonal complementary space (rangeF*p)⊥ to rangeF*p in the tangent space TF(p)N of N. Thus the tangent space TF(p)N of N for p ∈ M has the following decomposition:

Now, a smooth map F : (Mm , gM) → (Nn, gN) is called a Riemannian map at p1 ∈ M is a linear isometry between the inner product spaces

and

p2 = F(p1). Therefore, Fischer stated in [117] that a Riemannian map is a map that is as isometric as it can be. In other words, F* satisfies the equation

for X, Y . It follows that isometric immersions and Riemannian submersions are particular Riemannian maps with kerF* = {0} and (rangeF*)⊥ = {0}. It is known that a Riemannian map is a subimmersion, which implies that the rank of the linear map F*p : Tp M → TF(p)N is constant for p in each connected component of M [F² = rankF which is a bridge between geometric optics and physical optics. Since the left-hand side of this equation is continuous on the Riemannian manifold M and since rankF is an integer-valued function, this equality implies that rankF is locally constant and globally constant on connected components. Thus if M is quantized to integer and half-integer values. The eikonal equation of geometrical optics is solved by using Cauchy’s method of characteristics, whereby, for real valued functions FdF ² = 1 are obtained by solving the system of ordinary differential equations x′ = grad f (x). Since harmonic maps generalize geodesics, harmonic maps could be used to solve the generalized eikonal equation [117].

In [117], Fischer also proposed an approach to build a quantum model and he pointed out that the success of such a program of building a quantum model of nature using Riemannian maps would provide an interesting relationship between Riemannian maps, harmonic maps, and Lagrangian field theory on the mathematical side, and Maxwell’s equation, Schrödinger’s equation, and their proposed generalization on the physical side.

Riemannian maps between semi-Riemannian manifolds have been defined in [122] by putting in some regularity conditions. On the other hand, affine Riemannian maps were also investigated and some interesting decomposition theorems were obtained by using the existence of Riemannian maps. Moreover curvature relations for Riemannian maps were obtained in [123]. For Riemannian maps and their applications in spacetime geometry, see [122].

Since Riemannian maps include isometric immersions and Riemannian submersions as subclasses, starting from [241], we introduce and study new Riemannian maps as generalizations of holomorphic submanifolds, totally real submanifolds, CR-submanifolds, slant submanifolds, holomorphic submersions, anti-invariant submersions, semi-invariant submersions, and slant submersions. Since then a number of papers published on the Riemannian submersions and Riemannian maps ([218], [222], [243], [244], [247], [250], [251], [254] and [260]) have demanded the publication of this volume as an update on submersion theory and Riemannian maps.

As we have seen from previous paragraphs, although submanifolds of complex manifolds have been an active field of study for many years, there are few books ([111] and [311]) on the geometry of Riemannian submersions defined on complex manifolds. Moreover, these books only cover holomorphic submersions. The objective of this book is to focus on all new geometric results on Riemannian submersion theory and Riemannian maps with proofs and their applications, and to bring the reader to the frontiers of active research on related topics.

The book consists of six chapters. Chapter 1 covers preliminaries followed by up-to-date mathematical results in Chapters 4 on Riemannian maps. Chapter 2 is focused on applications of Riemannian submersions in active ongoing research area in Robotic theory and their brief applications in Kaluza-Klein theory. First, we deal with the geometry of certain matrix Lie groups and relate them with a significant work of Altafini [12] on forward kinematic maps.

In Chapter 4, we study the geometry of Riemannian maps and introduce some new notions (umbilical Riemannian map, pseudo-umbilical Riemannian map) for Riemannian maps. We also give a new Bochner identity and, by using this identity we obtain divergence theorem for a map. We also show that it is possible to obtain new totally geodesic conditions by using this divergence theorem. Moreover, some well-known results of isometric immersions have been carried into the geometry of Riemannian maps, like the Chen inequality, circle theorems, and Clairaut conditions.

The motivation of the rest of the chapters comes from the development of the general theory of Cauchy-Riemann (CR) submanifolds [26] and slant submanifolds [70], and it mainly contains a series of results obtained by the author and collaborators in recent years.

In Chapter 3, we study invariant Riemannian submersions, anti-invariant Riemannian submersions, semi-invariant Riemannian submersions, slant submersions, semi-slant submersions, generic submersions, hemi-slant submersions, and pointwise slant submersions from almost Hermitian manifolds onto arbitrary Riemannian manifolds. We show that these new submersions produce new conditions for submersions to be harmonic and totally geodesic. We give several examples and show that such submersions have rich geometric properties.

In Chapter 5, we study new Riemannian maps from almost Hermitian manifolds to Riemannian manifolds as generalizations of invariant Riemannian submersions, antiinvariant Riemannian submersions, semi-invariant submersions, generic submersions, slant submersions, semi-slant submersions, and hemi-slant submersions, and we give many examples and study the effect of such maps on the geometry of the domain manifold and the target manifold.

In Chapter 6, we study Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as generalizations of holomorphic submanifolds, totally real submanifolds, CR-submanifolds, generic submanifolds, slant submanifolds, semi-slant submanifolds, and hemi-slant submanifolds. We obtain many new properties of the domain manifolds and the target manifolds, and investigate the harmonicity and totally geodesic conditions for such maps. We also relate these maps with pseudo-harmonic maps.

The results included in this book should stimulate future research on Riemannian submersions and Riemannian maps. To the best of our knowledge, there does not exist any other book covering the material in this volume. Our approach, in this book, has the following special features:

• The first chapter of the book has been also designed to incorporate the latest developments in the Riemann geometry.

• Extensive list of cited references on the Riemannian geometry is provided for the readers to understand easily the main focus on the Riemannian submersions and Riemannian maps and their applications.

• Each chapter begins with a brief introduction to the background and the motivation of the topics under consideration. This will help readers grasp the main ideas more easily.

• There is an extensive subject index.

• The sequence of chapters is arranged so that the understanding of a chapter stimulates interest in reading the next one and so on.

• Applications are discussed separately (see Chapter 2) from the mathematical theory.

• Overall, the presentation is self-contained, fairly accessible, and, in some special cases supported by references.

This book is intended for graduate students and researchers who have good knowledge of Riemannian geometry and its submanifolds, and, interest in Riemannian submersions and Riemannian maps.

About the Author

Bayram Şahin is Professor of Mathematics in the Department of Mathematics at Ege University of Izmir at Turkey. Prior to coming to Ege University, he was a full professor in the Department of Mathematics at Inonu University of Malatya at Turkey. After he obtained his Ph.D. (Differential Geometry) from the University of Inonu, he spent 12 months at the University of Windsor, Windsor, Ontario, Canada as a Postdoctorate and a visiting research scholar. He is editorial board member for the Turkish Journal of Mathematics and Mediterranean Journal of Mathematics. He is the co-author of the book Differential Geometry of Lightlike Submanifolds published in 2010 by Springer-Verlag and he has written more than 80 articles on manifolds, submanifolds and maps between them in various peer-reviewed publications, including (among others) Geometriae Dedicata, Acta Applicandae Mathematica, International Journal of Geometric Methods in Modern Physics, Indagationes Mathematica, Central Europen Journal of Mathematics, Turkish Journal of Mathematics, Journal of the Korean Mathematical Society, Mediterranean Journal of Mathematics, Chaos, Solitons and Fractals.

Chapter 1

Basic Geometric Structures on Manifolds

Abstract

In this chapter, we give brief information about geometric structures which will be used in the following chapters. In addition to the basic concepts, some geometric notions such as symmetry conditions, parallelity conditions, new product structures on manifolds, generalized Einstein manifolds, and biharmonic maps introduced very recently will be presented in this chapter. The chapter consists of six sections. In the first section, the main notions and theorems of Riemannian geometry are reviewed, such as the Riemannian manifold, Riemannian metric, Riemannian connection, curvatures (Riemannian, sectional Ricci, Scalar), derivatives (exterior, covariant, inner), operators (Hessian, divergence, Laplacian), Einstein manifolds and their generalizations, symmetry conditions, and very brief notions from integration. In the second section, we look at vector bundles and related notions. In the third section, we recall basic notions from submanifold theory and distributions on manifolds. In the fourth section, we mention Riemannian submersions, O’Neill’s tensor fields, and curvature relations of Riemannian submersions. In this section, we also recall the notion of horizontally weakly conformal maps, which will be a crucial tool for a characterization of harmonic maps. In the fifth section, we study various product structures including warped product, twisted product, oblique warped product, and convolution product on Riemannian manifolds. We also provide connection relations and curvature expressions of each case. In the last section, we give a general setting for a map between manifolds such as a vector field along a map, a connection along a map, a curvature tensor field along a map, a second fundamental form of a map, a tension field of a map. We also recall harmonic maps and biharmonic maps in detail.

Keywords

Riemannian manifold; Riemannin connection; Riemannian curvature tensor; locally symmetric manifold; semisymmetric Riemannian manifold; pseudosymmetric manifold; Chaki-type pseudosymmetric manifold; Ricci tensor; Scalar curvature; Hesian; Ricci soliton; gradient Ricci soliton; gradient Ricci almost soliton; quasi-Einstein manifold; generalized quasi-Einstein manifold; quasi-Einstein Riemannian manifold; generalized m– quasi-Einstein manifold; divergence; Laplacian; vector bundle; pullback bundle; pullback connection; Riemannian submanifold; Gauss formula; Weingarten formula; parallel submanifold; semi-parallel submanifold; pseudo-parallel submanifold; Chen inequality; distribution; Riemannian submersion; O’Neill’s tensors; weakly conformal maps; horizontally weakly conformal maps; usual product; warped product; multiply warped product; doubly warped product; twisted product; multiply twisted product; doubly twisted product; D-homothetic warping; oblique warped product; convolution product; second fundamental form; harmonic map; biharmonic map; tension field; bi-tension field

Nothing in this world can take the place of persistence. Talent will not; nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not; the world is full of educated derelicts. Persistence and determination alone are omnipotent

Calvin Coolidge

Contents

1. Riemannian manifolds and related topics 

2. Vector bundles 

3. Riemannian submanifolds and distributions 

4. Riemannian submersions 

5. Certain product structures on manifolds 

6. Geometric structures along a map 

1 Riemannian manifolds and related topics

In this section, we set out the basic materials needed in this book. We begin by reviewing some results of Riemannian geometry.

Let M be a real n-dimensional smooth manifold. First of all, we emphasize that throughout this book, all manifolds and structures on (between) them are supposed to be differentiable of class C∞. We also note that C∞(M) and χ(M) are, respectively, the algebra of differentiable functions on M and the module of differentiable vector fields of M. A linear connection on M is a map

such that

for arbitrary vector fields X, Y, Z and smooth functions f, h on M. ∇X is called the covariant derivative operator and ∇XY is called the covariant derivative of Y with respect to X. Define a tensor field ∇Y, of type (1, 1), and given by (∇Y)(X) = ∇XY, for any Y. Also, ∇Xf = X f is the covariant derivative of f along X. The covariant derivative of a