This book provides a striking synthesis of the standard theory of connections in principal bundles and the Lie theory of Lie groupoids. The concept of Lie groupoid is a little-known formulation of the concept of principal bundle and corresponding to the Lie algebra of a Lie group is the concept of Lie algebroid: in principal bundle terms this is the Atiyah sequence. The author's viewpoint is that certain deep problems in connection theory are best addressed by groupoid and Lie algebroid methods. After preliminary chapters on topological groupoids, the author gives the first unified and detailed account of the theory of Lie groupoids and Lie algebroids. He then applies this theory to the cohomology of Lie algebroids, re-interpreting connection theory in cohomological terms, and giving criteria for the existence of (not necessarily Riemannian) connections with prescribed curvature form. This material, presented in the last two chapters, is work of the author published here for the first time. This book will be of interest to differential geometers working in general connection theory and to researchers in theoretical physics and other fields who make use of connection theory.
Author(s): K. Mackenzie
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1987
Language: English
Pages: 344
Cover......Page 1
London Mathematical Society Lecture Note Series. 124......Page 2
Lie Groupoids and Lie Algebroids in Differential Geometry......Page 4
9780521348829......Page 5
Contents......Page 6
Introduction......Page 8
1. Groupoids......Page 18
2. Morphisms, subgroupoids and quotient groupoids......Page 23
3. Transitive and totally intransitive groupoids......Page 30
CHAPTER II - Topological groupoids......Page 33
1. Basic definitions and examples......Page 34
2. Local triviality......Page 48
3. Components in topological groupoids......Page 61
4. Representations of topological groupoids......Page 66
5. Admissible sections......Page 74
6. The monodromy groupoid of a locally trivial groupoid......Page 80
7. Path connections in topological groupoids......Page 91
CHAPTER III - Lie groupoids and Lie algebroids......Page 99
1. Differentiable and Lie groupoids......Page 100
2. Lie algebroids......Page 117
3. The Lie algebroid of a differentiable groupoid......Page 126
4. The exponential map and adjoint formulas......Page 142
5. Infinitesimal connection theory and the concept of transition form......Page 156
6. The Lie theory of Lie groupoids over a fixed base......Page 174
7. Path connections in Lie groupoids......Page 184
CHAPTER IV - The cohomology of Lie algebroids......Page 202
1. The abstract theory of transitive Lie algebroids......Page 204
2. The cohomology of Lie algebroids......Page 214
3. Non-abelian extensions of Lie algebrolds and the existence of transitive Lie algebroids with prescribed curvature......Page 229
4. The existence of local flat connections and families of transition forms......Page 250
5. The spectral sequence of a transitive Lie algebroid......Page 258
CHAPTER V - An obstruction to the Integrability of transitive Lie algebroids......Page 276
1. Results......Page 277
2. Epilogue......Page 285
APPENDIX A - On principal bundles and Atiyah sequences......Page 289
1. Principal and fibre bundles......Page 290
2. Quotients of vector bundles over group actions......Page 294
3. The Atiyah sequence of a principal bundle......Page 299
4. Infinitesimal connections and curvature......Page 308
1. Definitions and notations......Page 325
2. Formulas for the right derivative......Page 327
APPENDIX C - On vector bundles......Page 330
REFERENCES......Page 334
INDEX......Page 340
