Introduction .................................................... 1
I. Basic, notions
1. Differential calculus ................................. 14
2. Manifolds ............................................. 20
3. Tangent bundle and general fiber bundles .............. 22
4. The Lie bracket of vector fields ...................... 25
5. Lie groups and symmetric spaces: basic facts .......... 30
II. Interpretation of tangent objects via scalar extensions
6. Scalar extensions. I: Tangent functor and dual
numbers ............................................... 36
7. Scalar extensions. II: Higher order tangent
functors .............................................. 42
8. Scalar extensions. Ill: Jet functor and truncated
polynomial rings ...................................... 50
III. Second order differential geometry
9. The structure of the tangent bundle of a vector
bundle ................................................ 57
10. Linear connections. I: Linear structures on bilinear
bundles ............................................... 61
11. Linear connections. II: Sprays ........................ 68
12. Linear connections. Ill: Covariant derivative ......... 71
13. Natural operations. I: Exterior derivative of a
one-form .............................................. 73
14. Natural operations. II: The Lie bracket revisited ..... 75
IV. Third and higher order differential geometry
15. The structure of TkF: Multilinear bundles ............. 79
16. The structure of TkF: Multilinear connections ......... 83
17. Construction of multilinear connections ............... 87
18. Curvature ............................................. 91
19. Linear structures on jet bundles ...................... 95
20. Shifts and symmetrization ............................. 98
21. Remarks on differential operators and symbols ........ 102
22. The exterior derivative .............................. 106
V. Lie Theory
23. The three canonical connections of a Lie group ....... 110
24. The structure of higher order tangent groups ......... 116
25. Exponential map and Campbell-Hausdorff formula ....... 124
26. The canonical connection of a symmetric space ........ 128
27. The higher order tangent structure of symmetric
spaces ............................................... 134
VI. Diffeomorphism Groups and the exponential jet
28. Group structure on the space of sections of TkM ...... 139
29. The exponential jet for vector fields ................ 144
30. The exponential jet of a symmetric space ............. 148
31. Remarks on the exponential jet of a general
connection ........................................... 151
32. From germs to jets and from jets to germs ............ 153
Appendix L. Limitations ....................................... 156
Appendix G. Generalizations ................................... 159
Appendix: Multilinear Geometry
BA. Bilinear algebra .................................... 161
MA. Multilinear algebra ................................. 168
SA. Symmetric and shift invariant multilinear algebra ... 182
PG. Polynomial groups ................................... 192
References .................................................... 199
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