English
norskBlar i forfatter "Goldberg, Vladislav V."
Viser treff 1-8 av 8
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Abelian equations and rank problems for planar webs
(Journal article; Tidsskriftartikkel; Peer reviewed, 2006-05-04)We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincar´e’s theorem: a planar 4- web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs ... -
Geodesic Webs and PDE Systems of Euler Equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2008-10-30)We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x1, ..., xn) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d ≥ n + 1) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler ... -
Geodesic Webs of Hypersurfaces
(Working paper; Arbeidsnotat, 2008-12-11)In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n + 2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provide that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all ... -
Geodesic Webs on a Two-Dimensional Manifold and Euler Equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2008-10-30)We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the web foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar ... -
Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds
(Journal article; Tidsskriftartikkel; Peer reviewed, 2004-03-31)We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f(x, y) and g4(x, y), ..., gd(x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order ... -
On a class of linearizable planar geodesic webs
(Working paper; Arbeidsnotat, 2008-12-13)We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb. -
On Rank Problems for Planar Webs and Projective Structures
(Chapter; Bokkapittel, 2008-12-03)We present some old and recent results on rank problems and linearizability of geodesic planar webs -
On the Blaschke Conjecture for 3-Webs
(Journal article; Tidsskriftartikkel; Peer reviewed, 2004-11-21)We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conjecture does not exceed fifteen and give criteria for rigidity of 3-webs.
