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FOUNDATIONS

OF DIFFERENTIAL

GEOMETRY

Peter W. Michor

Mailing address: Peter W. Michor, Institut f"ur Mathematik der Universit"at Wien,

Strudlhofgasse 4, A-1090 Wien, Austria.

E-mail [email protected]

These notes are from a lecture course

Differentialgeometrie und Lie Gruppen which has been held at the University of Vienna during the academic year 1990/91, again in 1994/95, and in WS 1997. It is not yet complete and will be enlarged during the year.

In this lecture course I give complete definitions of manifolds in the beginning, but (beside spheres) examples are treated extensively only later when the theory is developed enough. I advise every novice to the field to read the excellent lecture notes

Typeset by AMS-TEX

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TABLE OF CONTENTS 1. Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . 1 2. Submersions and Immersions . . . . . . . . . . . . . . . . . . 13 3. Vector Fields and Flows . . . . . . . . . . . . . . . . . . . . 18 4. Lie Groups I . . . . . . . . . . . . . . . . . . . . . . . . . 39 5. Lie Groups II. Lie Subgroups and Homogeneous Spaces . . . . . . . 55 6. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . 64 7. Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 76 8. Integration on Manifolds . . . . . . . . . . . . . . . . . . . . 84 9. De Rham cohomology . . . . . . . . . . . . . . . . . . . . . 91 10. Cohomology with compact supports

and Poincar'e duality . . . . . . . . . . . . . . . . . . . . . . 102 11. De Rham cohomology of compact manifolds . . . . . . . . . . . 117 12. Lie groups III. Analysis on Lie groups . . . . . . . . . . . . . . 124 13. Derivations

on the Algebra of Differential Forms and the Fr"olicher-Nijenhuis Bracket . . . . . . . . . . . . . . . 137 14. Fiber Bundles and Connections . . . . . . . . . . . . . . . . . 146 15. Principal Fiber Bundles and G-Bundles . . . . . . . . . . . . . 157 16. Principal and Induced Connections . . . . . . . . . . . . . . . 175 17. Characteristic classes . . . . . . . . . . . . . . . . . . . . . 196 18. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 222 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Draft from November 17, 1997 Peter W. Michor,

1 1. Differentiable Manifolds

1.1. Manifolds. A topological manifold is a separable metrizable space M which is locally homeomorphic to Rn . So for any x 2 M there is some homeomorphism u : U ! u(U ) ` Rn , where U is an open neighborhood of x in M and u(U ) is an open subset in Rn . The pair (U; u) is called a chart on M .