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Natural operators of smooth mappings of manifoldswith metric fields
Authors:Pavla Musilová  Jana Musilová
Institution:Institute of Theoretical Physics and Astrophysics, Masaryk University Brno, Kotlá?ská 2, 611 37 Brno, Czech Republic
Abstract:In this paper we introduce the concepts of both a natural bundle and a natural operator generalized for the case of the category Mfm × Mfm of cartesian products of two manifolds and products of local diffeomorphisms. It is shown that any r-th order natural bundle over M × N has a structure of an associated bundle (PrM × PrN)Z Gmr × Gmr]. We consider prolongations of such associated bundles and their reduction with respect to a chosen subgroup. The existence of a bijective correspondence between natural operators of order k and the equivariant mappings of the corresponding type fibers are proved. A basis of invariants of arbitrary order is constructed for natural operators of smooth mappings of manifolds endowed with metric fields or connections, with values in a natural bundle of order one.
Keywords:natural bundle  natural operator  orbit reduction method  covariant differential
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