Spaces over algebras and their applications
This is a survey paper related to some aspects studied in the Kazan school of differential geometry about the geometric structures defined by some real algebras. In fact the first results about the relationship between the dual algebras and the screw geometry of the 3-dimensional Euclidean space have been obtained by A. P. Kotel'nikov in 1895. Then D. N. Zeiliger developed Kotel'nikov's ideas by using the transfer method in the ruled differential geometry. Next there are the results of A. P. Norden related to the bi-affine and bi-axial spaces. Spaces over algebras have been studied by B. A. Rozenfel'd, G. I. Kryuchkovich, V. V. Vishnevskij, P. A. Shirokov, the author of the present survey paper and others. The author presents the main algebraic aspects about regular representations of algebras and the corresponding spaces over algebras and the corresponding generalizations. Then he presents certain applications of the spaces over algebras in differential geometry. The bundle of \(T^k(M_n)\) of velocities of order \(k\) on the smooth manifold \(M_n\) is related to the algebra \({\mathbb{R}}({\epsilon}^k)\) of plural numbers of order \(k\), generated by \(1\) and \(\epsilon\) with the property \({\epsilon}^{k+1}=0\). The main results are concerned with the idea of vertical, complete and natural lifts. Next one presents some results about the relationship between the differential geometry of hypersurfaces in affine spaces and the algebra of dual numbers. The paper contains a bibliography consisting 161 titles of papers and books written, mainly by Russian mathematicians (most of them being from Kazan). The paper contains several useful informations, however they are quite partial.
scientific article; zbMATH DE number 1768744