Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

Abstract: Motivated by the theory of Painleve equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stackel systems.

Three-dimensional cr-submanifolds in the nearly kahler 6-sphere with one-dimensional nullity

Abstract: In this paper, we study certain three-dimensional CR-submanifolds M of the nearly Kahler 6-dimensional sphere S6(1). It is well known that there does not exist a three-dimensional totally geodesic proper CR-submanifold in S6(1). In this paper we obtain a classification of the 3-dimensional CR-submanifolds which are the closest possible to totally geodesic submanifolds, i.e. those that admit a one-dimensional nullity distribution.

The generalized Taub-NUT congruence in Minkowski spaces

Abstract: With any shear-free congruence of null geodesics in a Lorentzian geometry there is associated a Cauchy-Riemann three-space; and in certain spacetimes including the Ricci-flat spacetimes with expanding null shear-free (n.s.f.) congruences the deviation form of the congruence picks out an integrable distribution of complex two-spaces in the CR geometry. Conversely, given a CR geometry with an integrable distribution of two-spaces one can construct an associated family of spacetimes with a null, shear-free congruence. The interesting problem is the restrictionR ab =0. We consider the case of n.s.f. congruences in Minkowski spacetime constructed from CR geometries of maximal symmetry. The special two-spaces are here taken to be those associated with either the Taub-NUT geometry or, as a limiting case, those associated with the Hauser twisting typeN solution. We obtain the most general solution for these cases.

Integrability of exit times and ballisticity for random walks in Dirichlet environment

Abstract: We consider random walks in Dirichlet environment, introduced by Enriquez and Sabot in 2006. As this distribution on environments is not uniformly elliptic, the annealed integrability of exit times out of a given finite subset is a non-trivial property. We provide here an explicit equivalent condition for this integrability to happen, on general directed graphs. Such integrability problems arise for instance from the definition of Kalikow auxiliary random walk. Using our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot.

Viability theorem for deterministic mean field type control systems

Abstract: A mean field type control system is a dynamical system in the Wasserstein space describing an evolution of a large population of agents with mean-field interaction under a control of a unique decision maker. We develop the viability theorem for the mean field type control system. To this end we introduce a set of tangent elements to the given set of probabilities. Each tangent element is a distribution on the tangent bundle of the phase space. The viability theorem for mean field type control systems is formulated in the classical way: the given set of probabilities on phase space is viable if and only if the set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.