Browsing by Author "Malakhaltsev M."

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3-webs with singularities
(Maik Nauka-Interperiodica Publishing, 2016) Arias Amaya, Fabián; Arteaga Bejarano J.R.; Malakhaltsev M.
A 3-web with singularities is an ordered collection of three one-dimensional distributions L1, L2, L3 on a 2-dimensional manifold M. The subset Σ ⊂ M where these distributions are not pairwise transversal is called the singularity set. Under some conditions on Σ we find the differential invariants of the 3-web with singularities at the points of Σ and give examples of calculation of these invariants. © 2016, Pleiades Publishing, Ltd.
A generalization of the Gauss–Bonnet–Hopf–Poincaré formula for sections and branched sections of bundles
(Elsevier B.V., 2017) Arias Amaya, Fabián; Malakhaltsev M.
For a two-dimensional compact oriented Riemannian manifold (M,g), and a vector field V on M, the Hopf–Poincaré theorem combined with the Gauss–Bonnet theorem gives the Gauss–Bonnet–Hopf–Poincaré (GBHP) formula: ∑z∈Z(V)indz(V)= [Formula presented] ∫MKdσ, where Z(V) is the set of zeros of V, indz(V) is the index of V at z∈Z(V), and K is the curvature of g. We consider a locally trivial fiber bundle π:E→M over a compact oriented two-dimensional manifold M, and a section s of this bundle defined over M∖Σ, where Σ is a discrete subset of M called the set of singularities of the section. We assume that the behavior of the section s at the singularities is controlled in the following way: s(M∖Σ) coincides with the interior part of a surface S⊂E with boundary ∂S, and ∂S is π−1(Σ). For such sections s we define an index of s at a point of Σ, which generalizes in the natural way the index of zero of a vector field, and then prove that the sum of these indices at the points of Σ can be expressed as an integral over S of a 2-form constructed via a connection in E, thus we obtain a generalization of the GBHP formula. Also we consider branched sections with singularities, define an index of a branched section at a singular point, and find a generalization of the GBHP formula for the branched sections. © 2017
Topological Invariants of Principal G-Bundles with Singularities
(Pleiades Publishing, 2018) Arias Amaya, Fabián; Malakhaltsev M.
principal G-bundle with singularities is a principal bundle π: P¯ → M with structure group G¯ which reduces to a subgroup G ⊂ G¯ on the set M \ Σ, where M is an n-dimensional compact manifold and Σ ⊂ M is a k-dimensional submanifold. For example, a vector field on an n-dimensional Riemannian manifold M defines reduction of the orthonormal frame bundle of M to the subgroup O(n − 1) ⊂ O(n) on the set M \ Σ, where Σ is the set of zeros of this vector field. The aim of this paper is to construct topological invariants of principal bundles with singularities. To do this we apply the obstruction theory to the sectionM → P¯ /Gcorresponding to the reduction and obtain the topological invariant as a class in Hn−k(M,M \ Σ; πn−k−1(G¯ /G)). We study the properties of this invariants and, in particular, consider cases k = 0 y k = n − 1. © 2018, Pleiades Publishing, Ltd.