Initiation to Global Finslerian Geometry

265 Pages·2006·8.96 MB·English

by  Hassan Akbar-Zadeh Doctorat d Etat en Mathématiques Pures June 1961 La Sorbonne Paris.

Initiation to
Global Finslenan Geometry
North-Holland Mathematical Library
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VOLUME 68
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Initiation to Global
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v
PREFACE
This book is an initiation to global methods in the study of
the differential geometry of Finsler manifolds. It contains my
research on the subject during the last twenty years. Most of it has
been published in articles in different journals. I have brought it
all together in a streamlined manner to offer a coherent vision to
global differential Finslerian geometry.
The first three chapters form the foundation of Finslerian
geometry. They contain the basic notions of global Finslerian
geometry and lay the groundwork for the rest of the book. The
treatment is deliberately kept transparent and simple so as to
highlight the differences between Riemannian geometry and
Finslerian geometry.
At the start it is best to note that Finslerian geometry is the
most natural generalization of Riemannian geometry.
Consequently the results established in the book are the
generalizations of the Riemannian case. Our particular interest
bears on complete or compact manifolds. The book contains
detailed proofs of a certain number results that I have published in
the Comptes Rendus de l'Academie des Sciences de Paris.
The book has eight chapters. Each chapter begins with a
resume of the results contained in it. The number of publications
on Finslerian geometry is high. I indicate only a few of them in
the bibliography.
I want to thank here my friend Dr. Cyrille de Souza who has
helped me in the preparation of this book. My sincere thanks go to
Mr. Sevenster, editor of Elsevier, for his kindness, patience and
advice during the preparation of the manuscript. I am also thankful
to Ms. Andy Deelen, Administrative Editor, for her suggestions for
the final preparation of the text. Finally, I am grateful to Elsevier
for its interest in the progress of this branch of differential
geometry.
vVIi
Introduction
Finslerian geometry is the most natural generalization of
Riemannian geometry. In his dissertation (1854[1]) Riemann
already imagined a generalization of his metric. Then P.Finsler in
his thesis (1918[17]) generalized a certain number of theorems of
classical differential geometry. Berwald contributed to the
progress of this geometry [1928 [8]). But the connection
introduced by him is not Euclidean and deprives the Finslerian
geometry the simplicity and elegance of Riemannian geometry as
Elie Cartan remarks in his book (1933, [13]). Unfortunately some
scholars of Finslerain geometry have not paid heed to Cartan's
observation.
In his work Cartan studied the geometry of Finslerian
manifolds in the framework of metric manifolds with the help of a
Euclidean connection. He introduced the notion of the manifold of
line elements ([13],[14]) that is formed by a set of points and the
direction starting from those points. The parallel transport defined
by him preserves the length of vectors. With the introduction of
Euclidean connection in the neighbourhood of each linear element,
the manifold enjoys all the properties of a Euclidean manifold. In
other words the manifold is locally Euclidean.
The different infinitesimal connections introduced by Cartan
(linear, projective and conformal) can be dealt with in the same
geometric framework: C. Ehresmann published an article on this
topic with the title "infinitesimal connections in a differentiable
fibre bundle" (1950, pp. 29-55 [16]) in the context of any Lie
group provided a general framework for the introduction of
connections. A clear treatment of the subject was given by
Lichnerowicz in his "Theorie Globale des connexions et des
groupes d'holonomie"(1954 [27]). Thus the modern foundations
of Finslerian geometry are best laid in the framework of fibre
bundles as done in this book.
This book falls naturally into three parts:
IInnttrroodduucctitoionn vviiii
I. Basics of Finselrian Geometry (Chapters I and II)
II. Classification of Finslerian manifolds (Chapters IV, V, VI)
III. Isometries, Projective and Conformal Transformations
(Chapters III, VII, VIII)
viii
CONTENTS
Preface v
Introduction vi
Chapter I
Linear Connections on a Space of Linear Elements
Abstract
I. Regular Linear Connections
1. Fibre Bundles V(M) and W(M) 1
2. Frames and Co-frames 2
3. Tensors and Tensor forms 4
4. Linear connections 5
5. Absolute differential in a linear connection.
Regular linear connection 6
6. Exterior differential forms 9
II. Curvature and Torsion of a regular
linear connection
7. Torsion and curvature tensors of a general
linear connection
1. Torsion tensors 13
2. Curvature tensors 14
8. Particular case of a linear connection of directions.
Conditions of reduction 17
9. Ricci identities 18
10. Bianchi identities 20
11. Torsion and Curvature defined by a covariant derivation 21
Contents ix
Chapter II
Finslerian Manifolds
Abstract
1. Metric manifolds 23
2. Euclidean connections 24
3. The system of generators on W. 27
4. Special connections 30
5. Case of orthonormal frames and local coordinates
for the class of special connections 32
6. Finslerian manifolds 33
7. Finslerian connections 36
8. Curvature tensors of the Finslerian connection 40
9. Almost Euclidean connections 44
Chapter III
Isometries and affine vector fields on the unitary
tangent fibre bundle
Abstract
1. Local group of 1-parameter local transformations
and Lie derivative 49
2. Local invariant sections 53
3. Introduction of a regular linear connection 54
4. The Lie derivative of a tensor in the large sense 57
5. The Lie derivative of the coefficients of a regular
linear connection 58
6. Fundamental formula 61
7. Divergence formulas 64
8. Infinitesimal isometries, the compact case 67
9. Ricci curvatures and Infinitesimal isometries 70
10. Infinitesimal affine transformations 75
11. Affine infinitesimal transformations and Covariant
Derivations 76