I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index.

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Gauge-natural operators transforming connections to the tangent bundle

Natural operations in differential geometry

This paper discusses the global theory of differentially geometric objects.

Natural bundles and operators

Recently, Kola? has determined all gauge-natural operators for any Lie group G of curvature type transforming every principal connection into modified curvature operator, [3]. In the case the structure group is the general linear group GL(m,'R.) in an arbitrary dimension m he has obtained two parameter family of all GL(m, R)-natural operators of curvature type consisting of a curvature of connection and a contracted curvature of connection and generated by linear adjoint invariant maps of the Lie algebra gl(m, R) into itself of the form id and A >-+ tr A • id, [4]. Using a general method by Kolar, [3], [4], [5], we determine all gaugenatural operators for any Lie group G defined on the bundle Q © Q of pairs of principal connections with values in L ® ®2T*B. We deduce that all such operators form a system generated by two modified curvatures of both principal connections and by values on the difference of these connections of some map induced by bilinear adjoint invariant map of the Lie algebra g of G. In the case the structure group is the general linear group GL(m, R) in an arbitrary dimension m we obtain 10-parameter family of all GL(m, R)-natural operators of curvature type consisting 4-parameter system generated by curvatures and contracted curvatures both connections and 6-parameter system generated by values on the difference of these connections of some bilinear and adjoint invariant maps of the Lie algebra g/(m, R). The author is grateful to Professor I. Kolai for suggesting the problem, valuable remarks and useful discussions.

background

On gauge-natural operators of curvature type on pairs of connections

Let n;r;k be natural numbers such that n k + 1. Non-existence of natural operators C r 0 Q(regT r k ! K r k ) and C r 0 Q(regT r k ! K r k ) over n-manifolds is proved. Some generalizations are obtained.

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Gauge-natural operators transforming connections to the tangent bundle

Citations

Natural operations in differential geometry

Natural bundles and operators

On gauge-natural operators of curvature type on pairs of connections

Non-existence of some natural operators on connections

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