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NATURAL OPERATIONS

IN DIFFERENTIAL

GEOMETRY

Ivan Kol'a^r Peter W. Michor

Jan Slov'ak

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

Strudlhofgasse 4, A-1090 Wien, Austria.

Ivan Kol'a^r, Jan Slov'ak, Department of Algebra and Geometry Faculty of Science, Masaryk University Jan'a^ckovo n'am 2a, CS-662 95 Brno, Czechoslovakia

Springer-Verlag, 1993 Orders to: Springer-Verlag Heidelberg, Tiergartenstr. 17, D-69121 Heidelberg, Germany, tel. x-49-6221-487-0.

vi + 434 pages. Hardcover ISBN 3-540-56235-4, ISBN 0-387-56235-4. DM 138,-; L64,50; FF 520,-; Lit. 152.420; "oS 1076,40; sFr 121,50; U.S. Customers: For Price Information please contact SV New York (201)348- 4033 (Toll Free: 1 (800)SPRINGER)

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This is a shortened version containing only the table of contents, the preface, and the introductions to all chapters, and the index.

Typeset by AMS-TEX

v TABLE OF CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I.

MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4 1. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4 2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11 3. Vector fields and flows . . . . . . . . . . . . . . . . . . . . . 16 4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41 CHAPTER II.

DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . 49 6. Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 49 7. Differential forms . . . . . . . . . . . . . . . . . . . . . . . 61 8. Derivations on the algebra of differential forms

and the Fr"olicher-Nijenhuis bracket . . . . . . . . . . . . . . . 67 CHAPTER III.

BUNDLES AND CONNECTIONS . . . . . . . . . . . . . . . 76 9. General fiber bundles and connections . . . . . . . . . . . . . . 76 10. Principal fiber bundles and G-bundles . . . . . . . . . . . . . . 86 11. Principal and induced connections . . . . . . . . . . . . . . . 99 CHAPTER IV.

JETS AND NATURAL BUNDLES . . . . . . . . . . . . . . . 116 12. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 13. Jet groups . . . . . . . . . . . . . . . . . . . . . . . . . . 128 14. Natural bundles and operators . . . . . . . . . . . . . . . . . 138 15. Prolongations of principal fiber bundles . . . . . . . . . . . . . 149 16. Canonical differential forms . . . . . . . . . . . . . . . . . . 154 17. Connections and the absolute differentiation . . . . . . . . . . . 158 CHAPTER V.