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CHAPTER Ⅰ．INTRODUCTION 1

1．Motions in a Riemannian space 1

2．Affine motions in a space with a linear connexion 6

3．Lie derivatives of scalars,vectors and tensors 9

4．The Lie derivative of a linear connexion 15

CHAPTER Ⅱ．LIE DERIVATIVES OF GENERAL GEOMETRIC OBJECTS 18

1．Geometric objects 18

2．The Lie derivative of a geometric object 19

3．Miscellaneous examples of Lie derivatives 22

4．Some general formulas 24

CHAPTER Ⅲ．GROUPS OF TRANSFORMATIONS LEAVING A GEOMETRIC OBJECT INVARIANT 30

1．Projective and conformal motions 30

2．Invariance group of a geometric object 32

3．A group as invariance group of a geometric object 36

4．Generalizations of the preceding theorems 42

5．Some applications 45

CHAPTER Ⅳ．GROUPS OF MOTIONS IN Vn 48

1．Groups of motions 48

2．Groups of translations 50

3．Motions and affine motions 51

4．Some theorems on projectively or conformally related spaces 52

5．A theorem of Knebelman 54

6．Integrability conditions of Killing＇s equation 56

7．A group as group of motions 57

8．A theorem of Wang 60

9．Two theorems of Egorov 63

10．Vn＇s admitting a group Gr of motions of order r = ？n(n - 1) + 1 67

11．Case Ⅰ 75

12．Case Ⅱ 80

CHAPTER Ⅴ．GROUPS OF AFFINE MOTIONS 85

1．Groups of affine motions 85

2．Groups of affine motions in a space with absolute parallelism 86

3．Infinitesimal transformations which carry affine conics into affine conics 89

4．Some theorems on affine and projective motions 91

5．Integrability conditions of ？= 0 93

6．An Ln with absolute parallelism which admits a simply transitive group of particular affine motions 95

7．Semi-simple group space 98

8．A group as group of affine motions 101

9．Groups of affine motions in an Ln or an An 105

10．Ln＇s admitting an n2-parameter complete group of motions 111

11．An＇s which admit a group of affine motions leaving invariant a symmetric covariant tensor of valence 2 113

12．An＇s which admit a group of affine motions leaving invariant an alternating covariant tensor of valence 2 114

13．Groups of affine motions in an An of order greater than n2 - n + 5 118

CHAPTER Ⅵ．GROUPS OF PROJECTIVE MOTIONS 130

1．Groups of projective motions 130

2．Transformations carrying projective conies into projective conics 131

3．Integrability conditions of ？ = 2p(uA？) 133

4．A group as group of projective motions 135

5．The maximum order of a group of projective motions in an An with non vanishing projective curvature 138

6．An An admitting a complete group of affine motions of order greater than n2 — n + 1 149

7．An Ln admitting an n2-parameter group of affine motions 155

CHAPTER Ⅶ．GROUPS OF CONFORMAL MOTIONS 157

1．Groups of conformal motions 157

2．Transformations carrying conformal circles into conformal circles 158

3．Integrability conditions of ？ = 2φuλ 160

4．A group as group of conformal motions 164

5．Homothetic motions 166

6．Homothetic motions in conformally related spaces 170

7．Subgroups of homothetic motions contained in a group of conformal motions or in a group of affine motions 171

8．Integrability conditions of ？gμλ = 2cgμλ 173

9．A group as group of homothetic motions 174

CHAPTER Ⅷ．GROUPS OF TRANSFORMATIONS IN GENERALIZED SPACES 177

1．Finsler spaces 177

2．Lie derivative of the fundamental tensor 179

3．Motions in a Finsler space 180

4．Finsler spaces with completely integrable equations of Killing 182

5．General affine spaces of geodesics 185

6．Lie derivatives in a general affine space of geodesics 188

7．Affine motions in a general affine space of geodesies 190

8．Integrability conditions of the equations ？= 0 190

9．General projective spaces of geodesies 194

10．Projective motions in a general projective space of geodesies 199

11．Integrability conditions of ？ = ？ 201

12．Affine spaces of k-spreads 207

13．Projective spaces of k-spreads 211

CHAPTER Ⅸ．LIE DERIVATIVES IN A COMPACT ORIENTABLE RIEMANNIAN SPACE 214

1．Theorem of Green 214

2．Harmonic tensors 215

3．Lie derivative of a harmonic tensor 217

4．Motions in a compact orientable Vn 218

5．Affine motions in a compact orientable Vn 221

6．Symmetric Vn 222

7．Isotropy groups and holonomy groups 223

CHAPTER Ⅹ．LIE DERIVATIVES IN AN ALMOST COMPLEX SPACE 225

1．Almost complex spaces 225

2．Linear connexions in an almost complex space 228

3．Almost complex metric spaces 230

4．The curvature in a pseudo-K？hlerian space 233

5．Pseudo-analytic vectors 235

6．Pseudo-Kahlerian spaces of constant holomorphic curvature 238

BIBLIOGRAPHY 244

APPENDIX 263

BIBLIOGRAPHY 288

AUTHOR INDEX 295

SUBJECT INDEX 298