Prolongations generated by horizontal vectors
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The linear frame bundle over a smooth manifold is considered. It is shown that covariant derivatives of geometrical objects given on the linear frame bundle are the images of the horizontal frame vectors under maps defined by the differentials of the geometrical objects. In a (first-order) affine connection that is true for tensors, vertical and horizontal frame vectors, and arbitrary vectors from the tangent space to the linear frame bundle. For the very first-order affine connection object and non-vertical frame vectors this property holds in a prolongation of the affine connection. Prolongations of the horizontal frame vectors are constructed.
KeywordsTangent-valued forms second-order tangent space prolongation of affine connection covariant derivatives first- and second-order horizontal vectors
Mathematics Subject ClassificationPrimary 53B05 53C05 Secondary 58A10
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The author declares no conflict of interest.
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