УКОРОЧЕНІ ВІДОБРАЖЕННЯ ПРОСТОРІВ АФФІННОЇ ЗВ’ЯЗНОСТІ

Abstract

The long history of theory of mappings was revived thanks to the tensor methods of inquiry. The notion of affine connectivity was introduced a hundred years ago. It enabled us to look at classic geometric problems from a different angle. Following the common tradition, this paper introduces a notion of a mapping for a space of affine connectivity. Modifying the method of A. P. Norden, we found the formulae for the main tensors: deformation tensor, Riemann tensor, Ricci tensor and their first and second covariant derivatives for spaces and , which are connected by a given mapping. These formulae contain both objects of and with covariant derivatives in respect to relevant connectivities. In order to simplify the expression, we introduced the notion of shortened mapping and its particular case: a half-mapping. The connectivity that appears in the case of a half-mapping is called a medium connectivity. The above mentioned formulae can be notably simplified in the case of transition to covariant derivatives in the medium connectivity. This fact permits us to obtain characteristics (the necessary conditions) for the estimates whether an object of inner character from the space of affine connectivity is preserved under a given type of mappings. Objects of the inner character are geometric objects implied by an affine connectivity. They include Riemann tensor, Ricci tensor, Weyl tensor. Every type of mapping received its own set of differential equations in covariant derivatives, which define a deformation tensor of connectivity with a necessity. The study of these equations can proceed by a research on integrability conditions. Integrability conditions are algebraic over-defined systems. That’s why there is a constant need in introduction of additionally specialized spaces or certain objects of these spaces. Applying the method of N. S. Sinyukov and J. Mikes, in the case of certain algebraic conditions, we obtained a form of a deformation tensor for a given mapping. Let us note that the medium connectivity was selected in order to simplify the calculations. Depending on the type of a model under consideration or on the physical limitations, we can construct any other connectivity (and mappings), which would be better suited for the given conditions. This approach is particularly fruitful when applied for invariant transformations connecting pairs of spaces of affine connectivity via their deformation tensor of connectivity.