Paul Popescu

Poisson structures on almost complex Lie algebroids

In this paper, we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.

Contact structures on Lie algebroids

In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riemannain manifold.

Foliated vector bundles and Riemannian foliations

Abstract In this Note we prove the equivalence between the Riemannian foliation and each of the following conditions: 1) the lifted foliation F r on the bundle of r-transverse jets is Riemannian for r ⩾ 1 ; 2) the foliation F 0 r on the slashed J 0 r is Riemannian and vertically exact for r ⩾ 1 ; 3) there exists a positively admissible transverse Lagrangian on J 0 r E , the r-transverse slashed jet bundle of a foliated bundle E → M , for r ⩾ 1 .

On higher order geometry on anchored vector bundles

Some geometric objects of higher order concerning extensions, semi-sprays, connections and Lagrange metrics are constructed using an anchored vector bundle.

On associated quasi connections

The purpose of this paper is to give a necessary and sufficient condition on the existence of associated splittings (defined in this paper) and to consider some applications to associated quasi-connections on fibred manifolds and vector bundles, using the idea and extending Theorem 1 from [2]. In Section 1, a general condition on the existence of associated splittings is given. In Section 2, the basic constructions concerning q.c.s. used in the next Section are briefly described following [7]; they extend the q.c.s. of Wang [8, 1, 2]. In Section 3 there are proved two theorems on associated q.c.s. using essentially the main theorem from Section 1.

A Lagrangian form of tangent forms

Abstract The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dependent differential form on a tangent bundle. The action on curves of a tangent form is natural associated with that of a second order Lagrangian linear in accelerations, while the converse association is not unique. An equivalence relation of the tangent form, compatible with gauge equivalent Lagrangians, is considered. We express the Euler–Lagrange equation of the Lagrangian as a second order Lagrange derivative of a tangent form, considering controlled and higher order tangent forms. Hamiltonian forms of the dynamics generated are given, extending some quantization formulas given by Lukierski, Stichel and Zakrzewski. Using semi-sprays, local solutions of the E–L equations are given in some special particular cases.

ON GENERALIZED ALGEBROIDS

A generalized algebroid is defined in the paper. It includes the known definitions of Lie algebroid, prealgebroid and Courant algebroid, but the new definition of a generalized prealgebroid. A global groupoidlike structure is defined and a generalized algebroid is associated with. Some non-trivial examples are given. The morphisms of algebroids are defined. Contravariant functors from the categories of algebroids with an antisymmetric bracket in the category of graded differential algebras are constructed. It is proved that a generalized algebroid defines a Stefan-Sussmann foliation.

Holomorphic last multipliers on complex manifolds

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic last multipliers with real last multipliers. Also, we consider some applications of our study for holomorphic gradient vector fields on holomorphic Riemannain manifolds as well as for holomorphic Hamiltonian vector fields and holomorphic Poisson bivector fields on holomorphic Poisson manifolds.

On quasi connections on fibred manifolds

The purpose of this paper is to define a quasi-connection on a fibred manifold and its curvature. The constructions follow the ideas from some previous papers of the author [8, 9] where a nonlinear q.c. on a vector bundle and its curvature are defined. Some objects defined there (relative tangent spaces and almost Lie structures) are defined on some v.b.s defined here; they are used in the con struct ions or to give some new interpretations.

How many k-step linear block methods exist and which of them is the most efficient and simplest one?

Abstract There have appeared in the literature a lot of k -step block methods for solving initial-value problems. The methods consist in a set of k simultaneous multistep formulas over k non-overlapping intervals. A feature of block methods is that there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge–Kutta methods). All the formulas are usually obtained from a continuous approximation derived via interpolation and collocation at k + 1 points. Nevertheless, all the k -step block methods thus obtained may be considered as different formulations of one of them, which results to be the most efficient and simple formulation of all of them. The theoretical analysis and the numerical experiments presented support this claim.