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Complete lifts of tensor fields of type (1,k) to natural bundles. (English) Zbl 0547.55014
There is treated mainly the existence problem of a complete lift of a vector field on a differentiable manifold M into the Lie algebra of vector fields on a bundle space E of a bundle \(\pi\) : \(E\to M\). The definition of complete lift in the note agrees with what is given by K. Yano and S. Kobayashi, K. Yano and S. Ishihara and A. Morimoto when one specializes the bundle to the tangent bundle, tangent bundle of \(p^ r\)-velocities or bundle of infinitesimal near points. The complete lift in the note is defined for natural bundles, which turn out to be isomorphic to associated fibre bundles with the frame bundle over M.
After a study of the properties of the complete lift, there is given a necessary and sufficient condition for the existence of the complete lift for (1,k) tensor fields in the terminology of the invariance of a subspace in the Lie algebra associated to the tensor fields.
Reviewer: Y.Shikata

55R10 Fiber bundles in algebraic topology
58A20 Jets in global analysis
57R99 Differential topology
53C99 Global differential geometry