PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying (Fv2-a2)(Fv2+a2)(Fv2 - b2)(Fv2 + b2) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2n-dimensional Lagrange manifold E is defined and the F(±a2; ±b2)-structure on the vertical tangent space TV (E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on T(E). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r-parallel, r¯ anti half parallel when r = r¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

Collisionless Dynamics and the Cosmic Web

I review the nature of three-dimensional collapse in the Zeldovich approximation, how it relates to the underlying nature of the three-dimensional Lagrangian manifold and naturally gives rise to a hierarchical structure formation scenario that progresses through collapse from voids to pancakes, filaments and then halos. I then discuss how variations of the Zeldovich approximation (based on the gravitational or the velocity potential) have been used to define classifications of the cosmic large-scale structure into dynamically distinct parts. Finally, I turn to recent efforts to devise new approaches relying on tessellations of the Lagrangian manifold to follow the fine-grained dynamics of the dark matter fluid into the highly non-linear regime and both extract the maximum amount of information from existing simulations as well as devise new simulation techniques for cold collisionless dynamics.