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

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*Wolfgang Bertram*

#### 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