The Tao that can be told is not the eternal Tao.The name that can be named is not the eternal name.The nameless is the beginning of heaven and earth.The named is the mother often thousand things.Lao Tsu (Tao Te Ching[65],Ch.1) more special more general
Mathematics is the music of science and real analysis is the Bach of mathematics.There are many foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. Sterling K.Berberian[11]
1.The three concepts in the title lie at the heart of our book.The follow-manifolds withmanifoldsmanifolds withing diagram shows that the order in which they appear is also important:We note already at this point that by a connection we always mean a so-called Ehresmann connection.As a first approach,specifying an Ehresmann connection means the fixing of a direct summand of the(canonical) vertical subbundle of the double tangent bundle.Thus,it is a geometric notion,and indeed avery simple one.Our main goal was to give a comprehensive introduction to the theoryof Finsler manifolds,i.e.,manifolds endowed with a Finsler structure or Finsler function.The scheme above shows that the appropriate approach is the study of manifolds endowed with connections and sprays.Above all,we had in mind the needs of PhD students in Finsler geometry,and we tried to summarize the fundamentals of differential geometry together with a reasonable amount of the prerequisites from algebra and analysis in a single volume and in a coherent manner,and to expose the rudiments of Finsler geometry on these foundations.Experienced readers will notice that this approach makes it possible to derive several classical theorems from general principles in a simple and unified way. The tableofcontents comprises all the topics included in the book.In the following we only point out a few important features of our method. 2.As we have already mentioned,the first aim of our book is to serve as a textbook.To illustrate this,we note the following features:(1) In principle,the reading of the book requires only the knowledge of undergraduate linear algebra and analysis.However,readers with a background in classical differential geometry or elementary Riemannian ge-ometry will assimilate the material and perceive the subtleties more easily.(2) We define our technical terms unambiguously and use them in this spirit throughout.We state all our assertions in a clear and explicit manner,and we give detailed proofs for them.There are only a few exceptions to the latter,but in these few cases we nearly always give explicit references to the literature.In our proofs we carefully explain all steps,and we never leave non-trivial details to the reader(or atleast we do not intend to do so).(3) A course in‘Advanced Calculus'is an organic part of the book.Such a course could be formed from the relevant parts,arranged logically as follows:Append izA,Chapter 1,Appendix B,Append iC.1,Subsec-tions 4.2.1,3.2.1,Section 9.1.An on-standard feature of this course is the study of Finsler vector spaces,which is in close connection with convexity.Our teaching experience shows that the interest of upper undergraduate students in Finsler geometry maybe aroused in this way.
We note that Appendix C.2,C. 3 and Subsection 6.1.10 can easily be made independent of manifold theory,perhaps with some tutorial help.In this way we obtain an introduction to the classical theory of hypersurfaces of a finite-dimensional real vector space.3.The conceptual framework for the study of sprays and Finsler struc-tures is provided by the theory of manifolds and vector bundles.It must be clear from the foregoing that we do not explicitly relyon the reader's knowledge of manifold theory.However,our discussion of manifolds is rather concise,we usually restrict ourselves to the presentation of the most important concepts and facts,and we omit diffcult proofs.We do so not only due to the limited sizeof the book,but also because there are plenty of excellent textbooks on manifold theory,e.g., Barden and Thomas[9],Jef-frey M.Lee[66],John M.Lee[69],Mich or[76] and Tu[96],just to mention a few recent ones.There are also good course materials available on the In-ternet.Our main guides were such classical works as the first volume of the monograph of Greu b,Halperin and Vanstone[52] and‘Semi-Riemannian

Preface vii

Acknowledgments xiii

1Modules,Algebras and Derivations 1

1.1Modules and Vector Spaces 1

      1.1.1Basic Definitions and Facts 1

      1.1.2Homomorphisms 8

      1.1.3Cosets and Affine Mappings 18

1.2Tensors 20

      1.2.1Tensors as Multilinear Mappings 21

      1.2.2Substitution Operators and Pull-back 22

      1.2.3Canonical Isomorphisms 23

      1.2.4Tensor Components 24

      1.2.5Contraction and Trace 27

1.3Algebras and Derivations 30

      1.3.1Basic Definitions 30

      1.3.2Derivations 31

      1.3.3Lie Algebras 32

      1.3.4Graded Algebras and Graded Derivations 33

      1.3.5The Exterior Algebra of an R-module 35

      1.3.6Determinants 39

      1.3.7Volume Forms and Orientation 43

1.4Orthogonal Spaces 44

      1.4.1Scalar Product and Non-degeneracy 44

      1.4.2The Associated Quadratic Form 45

      1.4.3Orthonormal Bases 47

      1.4.4Orthogonal Mappings and the Adjoint 49

      1.4.5Modules with Scalar Product 51

2.Manifolds and Bundles 55

2.1Smooth Manifolds and Mappings 55

      2.1.1Charts,Atlases,Manifolds 55

      2.1.2Examples of Manifolds 57

      2.1.3Mappings of Class Cr 61

      2.1.4Smooth Partitions of Unity 67

2.2Fibre Bundles 68

      2.2.1Fibre Bundles,Bundle Maps,Sections 68

      2.2.2Vector Bundles 74

      2.2.3Examples and Constructions 82

      2.2.4Tt-tensors and T-tensor fields 89

      2.2.5Vector Bundles with Additional Structures 91

3.Vector Fields,Tensors and Integratuon 97

3.1Tangent Vectors and Tangent Space 97

      3.1.1Tangent Vectors and Tangent Space 97

      3.1.2The Derivative of a Differentiable Mapping 101

      3.1.3Some Local Properties of Differentiable Mappings 105

      3.1.4The Tangent Bundle of a Manifold 106

      3.1.5The Lie Algebra of Vector Fields 116

3.2First-order Differential Equations 122

      3.2.1Basic Existence and Uniqueness Theorems 122

      3.2.2Integral Curves 127

      3.2.3Flows 131

      3.2.4Commuting Flows 136

3.3Tensors and Differential Forms 141

      3.3.1The Cotangent Bundle of a Manifold 141

      3.3.2Tensors on a Manifold 144

      3.3.3Tensor Derivations 147

      3.3.4Differential Forms 151

      3.3.5The Classical Graded Derivations of A(M) 155

      3.3.6The Frolic her-Nijenhuis Theorem 160

3.4Integration on Manifolds 167

      3.4.1Orientable Manifolds 167

      3.4.2Integration of Top Forms 168

      3.4.3Stokes’Theorem 174

4.Structures on Tangent Bundles 179

4.1Vector Bundles on TM 179

      4.1.1Finsler Bundles and Finsler Tensor Fields 179

      4.1.2TheVectorBundleStructureofT:TTM→TM 188

      4.1.3The Vertical Subbundle of TTM 195

      4.1.4Acceleration and Re parametrizations 209

      4.1.5The Complete Lift of a Vector Field 211

      4.1.6The Vertical Endomorphism of TTM 220

      4.1.7Push-forwards 226

4.2Homogeneity 228

      4.2.1 Homogeneous Mappings of Vector Spaces228

      4.2.2Homogeneous Functions on TM 231

      4.2.3Homogeneous Vector Fields on TM 234

5.Sprays and Lagrangians 237

5.1Sprays and the Exponential Map 237

      5.1.1Second-order Vector Fields and Some of Their Mu-tants 237

      5.1.2Geodesics of a Semispray 247

      5.1.3The Exponential Map 253

      5.1.4The Theorem of Whitehead 257

5.2Lagrange Functions 262

      5.2.1Regularity and Global Dynamics 262

      5.2.2First Variation 268

6.Covariant Derivatives 277

6.1Differentiation in Vector Bundles 277

      6.1.1Covariant Derivative on a Vector Bundle 277

      6.1.2The Second Covariant Differential 287

      6.1.3Exterior Covariant Derivative 289

      6.1.4Metric Derivatives 291

      6.1.5Curvature and Torsion 293

      6.1.6The Levi-Civita Derivative 302

      6.1.7Covariant Derivative Along a Curve 307

      6.1.8Parallel Translation with Respect to a Covariant Derivative 311

      6.1.9Geodesics of an Aff n ely Connected Manifold 318

      6.1.10Hypersurfaces in a Riemannian Manifold 321

6.2Covariant Derivatives on a Finsler Bundle326

      6.2.1Curvature and Torsion 326

      6.2.2Deflection and Regularities 330

      6.2.3Vertical Covariant Derivative Operators 333

      6.2.4The Vertical Hessian of a Lagrangian 340

      6.2.5Parallelism and Geodesics 343

      6.2.6Metric u-covariant Derivatives 347

7.Theory of Ehresmann Connections 351

7.1Horizontal Subbundles 351

7.2Ehresmann Connections and Associated Objects 357

7.3Constructions of Ehresmann Connections 368

      7.3.1Ehresmann Connections and Projection Operators 368

      7.3.2Ehresmann Connections from Regular Covariant Derivatives 369

      7.3.3The Cramp in-Grif one Construction 371

7.4Some Useful Technicalities 374

7.5Homogeneity and Linearity 378

      7.5.1Homogeneity Conditions 378

      7.5.2The Ehresmann Connection of an Af finely Con-nected Manifold 383

      7.5.3The Linear Deviation 387

7.6Parallel Translation with Respect to an Ehresmann Con-nection 389

7.7Geodesics of an Ehresmann Connection 395

7.8Curvature and Torsion 396

7.9Ehresmann Connections and Covariant Derivatives 405

7.10The Induced Berwald Derivative 411

7.11The Debauch of Indices 416

7.12Tension,Torsion,Curvature and Geodesics Again 421

7.13The Berwald Curvature 426

7.14The A fine Curvature 431

7.15Linear Ehresmann Connections Revisited 439

8.Geometry of Spray Manifolds 445

8.1The Berwald Connection and Related Constructions 445

      8.1.1The Berwald Connection 445

      8.1.2The Induced Berwald Derivative 446

      8.1.3Torsion and Curvature 447

      8.1.4Coordinate Description 448

8.2Affine Deviation 451

      8.2.1The Jacobi Endomorphism 451

      8.2.2Jacobi Fields 460

8.3The Weyl Endomorphism 463

8.4Projective Changes 472

      8.4.1Project ively Related Sprays 472

      8.4.2Changes of Associated Objects 475

      8.4.3Project ively Related Covariant Derivatives 480

      8.4.4The Meaning of the Weyl Endomorphism 482

      8.4.5The Douglas Tensor 483

      8.4.6The Meaning of the Douglas Tensor 487

8.5Integrability and Flatness 494

9.Finsler Norms and Finsler Functions 503

9.1Finsler Vector Spaces 503

      9.1.1Convexity 503

      9.1.2Pre-Finsler Norms 512

      9.1.3Finsler Norms and Some of Their Characterizations 517

      9.1.4Reduction to Euclidean Vector Space 524

      9.1.5Averaged Scalar Product on a Gauge Vector Space 527

9.2Fundamentals on Finsler Functions 532

      9.2.1Pre-Finsler Manifolds 532

      9.2.2Finsler Functions and the Canonical Spray 539

      9.2.3The Rap csak Equations 543

      9.2.4Riemannian Finsler Functions 547

9.3Notable Covariant Derivatives on a Finsler Manifold 552

      9.3.1 The Fundamental Lemma of Finsler Geometry 552

      9.3.2The Finsler ian Berwald Derivative 555

      9.3.3The Cart an Derivative 563

      9.3.4The Chern-Rund and the Hashiguchi Derivative 564

9.4Isotropic Finsler Manifolds 568

      9.4.1Characterizations of Isotropy 568

      9.4.2The Flag Curvature 573

      9.4.3The Generalized Schur Theorem 575

9.5Geodesics and Distance 580

      9.5.1Finsler ian Geodesics and Isometries 580

      9.5.2The Finslerian Distance 585

      9.5.3The Myers-Steenrod Theorem 589

9.6Projective Relatedness Again 593

9.7Projective Me triz ability 597

9.8Berwald Manifolds 601

9.9Oriented Finsler Surfaces 608

      9.9.1Berwald Frames 608

      9.9.2The Fundamental Equations of Finsler Surfaces 610

      9.9.3Surviving Curvature Components 616

      9.9.4Szabo's‘Rigidity Theorem’ 619

Appendix A Sets,Mappings and Operations 625

A.1Set Notations and Concepts 625

A.2Mappings 627

A.3Groups and Group Actions 632

A.4Rings 640

Appendix B Topological Concepts 643

B.1Basic Definitions and Constructions 643

B.2Metric Topologies and the Contraction Principle 645

B.3More Topological Concepts 647

B.4Topological Vector Spaces 650

Appendix C Calculus in Vector Spaces 653

C.1Differentiation in Vector Spaces 653

C.2Canonical Constructions 665

      C.2.1Tangent Bundle and Derivative 669

      C.2.2Lifts of Functions 671

      C.2.3TheVectorBundleT:TTU→TU 671

      C.2.4Lifts of Vector Fields 673

      C.2.5 δ,i,j and J 674

C.3The Standard Covariant Derivative 676

Bibliography 681

General Conuentions 687

Notation Inder 689

Inder 695

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