Our aim is to show a method of finding all natural transformations of a functor into itself. We use here the terminology introduced in [4,5]. The notion of a soldered double linear morphism of soldered double vector spaces (fibrations) is defined. Differentiable maps commuting with -soldered automorphisms of a double vector space are investigated. On the set of such mappings, appropriate partial operations are introduced. The natural transformations are bijectively related with the elements...
A 3-web on a smooth -dimensional manifold can be regarded locally as a triple of integrable -distributions which are pairwise complementary, [5]; that is, we can work on the tangent bundle only. This approach enables us to describe a -web and its properties by invariant -tensor fields and where is a projector and id. The canonical Chern connection of a web-manifold can be introduced using this tensor fields, [1]. Our aim is to express the torsion tensor of the Chern connection through...
An anholonomic -web of dimension is considered as an -tuple of -dimensional distributions in general position. We investigate a family of -tensor fields (projectors and nilpotents associated with a web in a natural way) which will be used for characterization of all linear connections on a manifold preserving the given web.
Our aim is to find conditions under which a 3-web on a smooth -dimensional manifold is locally equivalent with a web formed by three systems of parallel -planes in . We will present here a new approach to this “classical” problem using projectors onto the distributions of tangent subspaces to the leaves of foliations forming the web.
Our aim is to demonstrate how the apparatus of groupoid terms (on two variables) might be employed for studying properties of parallelism in the so called -quasigroups. We show that an incidence structure associated with a medial quasigroup of type , , is either an affine space of dimension at least three, or a desarguesian plane. Conversely, if we start either with an affine space of order and dimension , or with a desarguesian affine plane of order then there is a medial quasigroup of...
In [19] we proved a theorem which shows how to find, under particular assumptions guaranteeing metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-less linear connection without flat points on a two-dimensional affine manifold. The result has the form of an implication only; if there are flat points, or if curvature is not recurrent, we have no good answer in general, which can be also demonstrated...
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