**Memoirs of the American Mathematical Society**

2008;
202 pp;
Softcover

MSC: Primary 22; 53; 58;
Secondary 14; 15; 51

Print ISBN: 978-0-8218-4091-7

Product Code: MEMO/192/900

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-0506-9
Product Code: MEMO/192/900.E**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

# Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

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*Wolfgang Bertram*

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

#### Table of Contents

# Table of Contents

## Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

- Contents v6 free
- Introduction 112 free
- I. Basic notions 1425 free
- II. Interpretation of tangent objects via scalar extensions 3647
- III. Second order differential geometry 5768
- 9. The structure of the tangent bundle of a vector bundle 5768
- 10. Linear connections. I: Linear structures on bilinear bundles 6172
- 11. Linear connections. II: Sprays 6879
- 12. Linear connections. Ill: Covariant derivative 7182
- 13. Natural operations. I: Exterior derivative of a one-form 7384
- 14. Natural operations. II: The Lie bracket revisited 7586

- IV. Third and higher order differential geometry 7990
- 15. The structure of T[sup(k)]F: Multilinear bundles 7990
- 16. The structure of T[sup(k)]F: Multilinear connections 8394
- 17. Construction of multilinear connections 8798
- 18. Curvature 91102
- 19. Linear structures on jet bundles 95106
- 20. Shifts and symmetrization 98109
- 21. Remarks on differential operators and symbols 102113
- 22. The exterior derivative 106117

- V. Lie Theory 110121
- VI.Diffeomorphism Groups and the exponential jet 139150
- Appendix L. Limitations 156167
- Appendix G. Generalizations 159170
- Appendix: Multilinear Geometry 161172
- References 199210