《The Theory of Lie Derivatives and Its Applications》

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CHAPTER Ⅰ．INTRODUCTION1

1．Motions in a Riemannian space1

2．Affine motions in a space with a linear connexion6

3．Lie derivatives of scalars,vectors and tensors9

4．The Lie derivative of a linear connexion15

CHAPTER Ⅱ．LIE DERIVATIVES OF GENERAL GEOMETRIC OBJECTS18

1．Geometric objects18

2．The Lie derivative of a geometric object19

3．Miscellaneous examples of Lie derivatives22

4．Some general formulas24

CHAPTER Ⅲ．GROUPS OF TRANSFORMATIONS LEAVING A GEOMETRIC OBJECT INVARIANT30

1．Projective and conformal motions30

2．Invariance group of a geometric object32

3．A group as invariance group of a geometric object36

4．Generalizations of the preceding theorems42

5．Some applications45

CHAPTER Ⅳ．GROUPS OF MOTIONS IN Vn48

1．Groups of motions48

2．Groups of translations50

3．Motions and affine motions51

4．Some theorems on projectively or conformally related spaces52

5．A theorem of Knebelman54

6．Integrability conditions of Killing＇s equation56

7．A group as group of motions57

8．A theorem of Wang60

9．Two theorems of Egorov63

10．Vn＇s admitting a group Gr of motions of order r = ？n(n - 1) + 167

11．Case Ⅰ75

12．Case Ⅱ80

CHAPTER Ⅴ．GROUPS OF AFFINE MOTIONS85

1．Groups of affine motions85

2．Groups of affine motions in a space with absolute parallelism86

3．Infinitesimal transformations which carry affine conics into affine conics89

4．Some theorems on affine and projective motions91

5．Integrability conditions of ？= 093

6．An Ln with absolute parallelism which admits a simply transitive group of particular affine motions95

7．Semi-simple group space98

8．A group as group of affine motions101

9．Groups of affine motions in an Ln or an An105

10．Ln＇s admitting an n2-parameter complete group of motions111

11．An＇s which admit a group of affine motions leaving invariant a symmetric covariant tensor of valence 2113

12．An＇s which admit a group of affine motions leaving invariant an alternating covariant tensor of valence 2114

13．Groups of affine motions in an An of order greater than n2 - n + 5118

CHAPTER Ⅵ．GROUPS OF PROJECTIVE MOTIONS130

1．Groups of projective motions130

2．Transformations carrying projective conies into projective conics131

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

4．A group as group of projective motions135

5．The maximum order of a group of projective motions in an An with non vanishing projective curvature138

6．An An admitting a complete group of affine motions of order greater than n2 — n + 1149

7．An Ln admitting an n2-parameter group of affine motions155

CHAPTER Ⅶ．GROUPS OF CONFORMAL MOTIONS157

1．Groups of conformal motions157

2．Transformations carrying conformal circles into conformal circles158

3．Integrability conditions of ？ = 2φuλ160

4．A group as group of conformal motions164

5．Homothetic motions166

6．Homothetic motions in conformally related spaces170

7．Subgroups of homothetic motions contained in a group of conformal motions or in a group of affine motions171

8．Integrability conditions of ？gμλ = 2cgμλ173

9．A group as group of homothetic motions174

CHAPTER Ⅷ．GROUPS OF TRANSFORMATIONS IN GENERALIZED SPACES177

1．Finsler spaces177

2．Lie derivative of the fundamental tensor179

3．Motions in a Finsler space180

4．Finsler spaces with completely integrable equations of Killing182

5．General affine spaces of geodesics185

6．Lie derivatives in a general affine space of geodesics188

7．Affine motions in a general affine space of geodesies190

8．Integrability conditions of the equations ？= 0190

9．General projective spaces of geodesies194

10．Projective motions in a general projective space of geodesies199

11．Integrability conditions of ？ = ？201

12．Affine spaces of k-spreads207

13．Projective spaces of k-spreads211

CHAPTER Ⅸ．LIE DERIVATIVES IN A COMPACT ORIENTABLE RIEMANNIAN SPACE214

1．Theorem of Green214

2．Harmonic tensors215

3．Lie derivative of a harmonic tensor217

4．Motions in a compact orientable Vn218

5．Affine motions in a compact orientable Vn221

6．Symmetric Vn222

7．Isotropy groups and holonomy groups223

CHAPTER Ⅹ．LIE DERIVATIVES IN AN ALMOST COMPLEX SPACE225

1．Almost complex spaces225

2．Linear connexions in an almost complex space228

3．Almost complex metric spaces230

4．The curvature in a pseudo-K？hlerian space233

5．Pseudo-analytic vectors235

6．Pseudo-Kahlerian spaces of constant holomorphic curvature238

BIBLIOGRAPHY244

APPENDIX263

BIBLIOGRAPHY288

AUTHOR INDEX295

SUBJECT INDEX298