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Boris S. Kruglikov

Publications and source records attributed to Boris S. Kruglikov.

8 recordsLinked to original sources
arXiv2022

Almost every path structure is not variational

Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian.

math.DG↗
arXiv2005

Dynimics and entropy in the Zhang model of Self-Organized Criticality

We give a detailed study of dynamical properties of the Zhang model, including evaluation of topological entropy and estimates for the Lyapunov exponents and the dimension of the attractor. In the thermodynamic limit the entropy goes to zero and the Lyapunov spectrum collapses.

math.DS↗
arXiv1997

On the Kobayashi-Royden pseudonorm for almost complex manifolds

In this paper we define Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. We prove that the pseudodistance induced by this pseudonorm coincides with the Kobayashi pseudodistance defined for the almost complex case earlier. We also consider a geometric application for moduli spaces of pseudoholomorphic curves.

dg-ga↗
arXiv1997

The Kobayashi pseudodistance on almost complex manifolds

We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familliar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.

dg-ga↗
arXiv1996

Nijenhuis tensors and obstructions for pseudoholomorphic mapping constructions

In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongations of the Cauchy-Riemann equation in 1-jets of the mapping from one almost complex manifold to another. We give a criterion of the prolongation existence of k-th pseudoholomorphic jets to the (k+1)-th ones. As a consequence basing on the Kuranishi's approach we obtain formal normal forms of almost complex structures. As another consequence we get the formal part of Newlander-Nirenberg's theorem on integrability. We also introduce the notion and study the properties of the linear Nijenhuis tensor, which we apply to almost complex structures of general position. Two applications are exhibited: a classification of four-dimensional almost complex structures in terms of distributions and a characterization of the new class of structures which we call Lie almost complex structures.

dg-ga↗
arXiv1996

Some classificational problems in four-dimensional geometry: distributions, almost complex structures and Monge-Ampere equations

In this paper we consider three deeply connected classificational problems on four-dimensional manifolds. First we consider and describe locally regular distributions. Second we give a classification of almost complex structures of general position in terms of distributions. Finally we classify nondegenerate Monge-Ampere equations with two variables in terms of e-structures.

dg-ga↗