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Affine analogues of the Sasaki-Shchepetilov connection

100%
Robaszewska M.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2016
|
tom 15
EN
For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.
2
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Criterion for connections on principal bundles over a pointed Riemann surface

100%
Biswas I.
Complex Manifolds
|
2017
|
tom 4
|
nr 1
155-171
EN
We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.
3
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Essential Killing fields of parabolic geometries: projective and conformal structures

84%
Čap A., Melnick K.
Open Mathematics
|
2013
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tom 11
|
nr 12
2053-2061
EN
We use the general theory developed in our article [Čap A., Melnick K., Essential Killing fields of parabolic geometries, Indiana Univ. Math. J. (in press)], in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.
4
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On geometry of curves of flags of constant type

84%
Doubrov B., Zelenko I.
Open Mathematics
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2012
|
tom 10
|
nr 5
1836-1871
EN
We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.
5
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The geometry of the space of Cauchy data of nonlinear PDEs

68%
Moreno G.
Open Mathematics
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2013
|
tom 11
|
nr 11
1960-1981
EN
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.
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