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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.

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$, Comment: 7 pages, no figures

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., Comment: 60 pages, 17 figures

Suppose the Riemannian metrics $g$ and $\bar g$ on a closed connected manifold $M^n$ are geodesically equivalent and strictly non-proportional at least at one point. Then the topological entropy of the geodesic flow of $g$ vanishes., Comment: This is a slightly extended version of the paper submitted to ETDS. 16 pages, one .eps figure

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., Comment: 14 pages, La-TeX

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., Comment: 19 pages, Latex

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., Comment: 43 pages, La-TeX, style , e-mail: lychagin@glas.apc.org, borkru@difgeo.math.msu.su and before 8th Dec 1996: lychagin@math.uit.no

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., Comment: 22 pages, La-Tex, style . (e-mail: lychagin@glas.apc.org, borkru@difgeo.math.msu.su and before 8th Dec 1996: lychagin@math.uit.no)