On reducibility of double linear connections on a double vector fibration with soldering
Medial quasigroups of type
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...
Parallelisability conditions for differentiable three-webs
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.
Double vector spaces
Connections on the second tangent bundle
Natural transformations of the second tangent functor and soldered morphisms
On geometry of the third tangent bundle
Some remarks on polynomial structures
Homomorphisms of projective planes over quasifields and nearfields
Places of alternative fields
On generalized formal power series
Construction of one epimorphism of projective planes over Cartesian groups
Double linear connections
Polynomial structures with double roots
Some algebraic models for differential geometry of second order [Abstract of thesis]
On metrizability of locally homogeneous affine 2-dimensional manifolds
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...
On Metrizable Locally Homogeneous Connections in Dimension
We discuss metrizability of locally homogeneous affine connections on affine 2-manifolds and give some partial answers, using the results from [Arias-Marco, T., Kowalski, O.: Classification of locally homogeneous affine connections with arbitrary torsion on 2-dimensional manifolds. Monatsh. Math. 153 (2008), 1–18.], [Kowalski, O., Opozda, B., Vlášek, Z.: A classification of locally homogeneous connections on 2-dimensional manifolds vis group-theoretical approach. CEJM 2, 1 (2004), 87–102.], [Vanžurová,...
Metrization problem for linear connections and holonomy algebras
We contribute to the following: given a manifold endowed with a linear connection, decide whether the connection arises from some metric tensor. Compatibility condition for a metric is given by a system of ordinary differential equations. Our aim is to emphasize the role of holonomy algebra in comparison with certain more classical approaches, and propose a possible application in the Calculus of Variations (for a particular type of second order system of ODE’s, which define geodesics of a linear...
Metrization of connections with regular curvature
We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also...
Soldered double linear morphisms
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...
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