Differential Forms

Ivan Kolář, Jan Slovák and Peter W. Michor
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Ivan Kolář: Masaryk University, Department of Algebra and Geometry, Faculty of Science
Jan Slovák: Masaryk University, Department of Algebra and Geometry, Faculty of Science
Peter W. Michor: Universität Wien, Institut für Mathematik

Chapter Chapter II in Natural Operations in Differential Geometry, 1993, pp 49-75 from Springer

Abstract: Abstract This chapter is still devoted to the fundamentals of differential geometry, but here the deviation from the standard presentations is already large. In the section on vector bundles we treat the Lie derivative for natural vector bundles, i.e. functors which associate vector bundles to manifolds and vector bundle homomorphisms to local diffeomorphisms. We give a formula for the Lie derivative of the form of a commutator, but it involves the tangent bundle of the vector bundle in question. So we also give a careful treatment to this situation. The Lie derivative will be discussed in detail in chapter XI; here it is presented in a somewhat special situation as an illustration of the categorical methods we are going to apply later on. It follows a standard presentation of differential forms and a thorough treatment of the Frölicher-Nijenhuis bracket via the study of all graded derivations of the algebra of differential forms. This bracket is a natural extension of the Lie bracket from vector fields to tangent bundle valued differential forms. We believe that this bracket is one of the basic structures of differential geometry (see also section 30), and in chapter III we will base nearly all treatment of curvature and the Bianchi identity on it.

Keywords: Vector Field; Vector Bundle; Differential Form; Tangent Bundle; Open Cover (search for similar items in EconPapers)
Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-02950-3_2

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DOI: 10.1007/978-3-662-02950-3_2

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