Weil bundle over the tensor product of two algebras of dual numbers

Background. Starting from the time of their discovery in 1953, Weil bundles have been actively studied by geometers of Russia, Japan, Czech Republic and other countries. The aim of this work is to construct the natural lifts of functions, 1-forms and vector fields from a base into Weil bundles over the tensor product of two algebras of dual numbers. Materials and methods. The methods of tensor algebra, theory of linear connections were used to achieve the objectives. Results. The authors constructed the tensor product of two algebras of dual numbers; the structural correlations of the algebra in a special basis, correlations of outer multiplication of linear forms on elements of the tensor product of two algebras of dual numbers were obtained; the description of the natural lifts of functions from a base into the studied Weil bundles was given. The natural lifts of vector fields, structural affinors for these Weil bundles were also introduced. It is shown how to obtain the vertical lifts of vector fields from the complete lift of a vector field with the help of available structural affinors. Finally, the natural lifts of 1-forms were constructed. Conclusions. The paper provides a summary of Weil bundles, natural extensions of functions from a base into a Weil bundle. The real-valued extensions of functions, vector fields and 1-forms from a base into a Weil bundle are described. The research results can be used to study the lifts of linear connections from a base into a Weil bundle over the tensor product of the algebras of dual numbers.