The aim of the paper is to prove that if some extended curvatures on a preinfinitesimal module L,... more The aim of the paper is to prove that if some extended curvatures on a preinfinitesimal module L, considered in the paper, vanish, then the derived preinfinitesimal module L(1) is a Lie pseudoalgebra. Two non-trivial examples are given. The first example is when L0 is an infinitesimal module and the second one is when L1 is a preinfinitesimal module.
The aim of the paper is to find conditions in which the derived almost Lie vector bundle E1 of an... more The aim of the paper is to find conditions in which the derived almost Lie vector bundle E1 of an almost Lie vector bundle E is a Lie algebroid. The conditions are that some extended curvatures on E, considered in the paper, are vanishing. Two non-trivial examples are given. One example is when E0 is a skew symmetric algebroid; the other one is when E1 is not a skew symmetric algebroid.
The aim of this paper is to construct two functors, called the derived functors of anchored modul... more The aim of this paper is to construct two functors, called the derived functors of anchored modules which allow linear connections. Some applications are given in the cases of vector bundles with differentials and Lie algebras. It is also shown that a linear connection and a skew-symmetric form on a preinfinitesimal module lift on the derived Lie pseudoalgebra to a null curvature connection and to a closed skew-symmetric form, respec-
The higher order bundles are considered using a vector pseudo-field on, in an inductive manner. T... more The higher order bundles are considered using a vector pseudo-field on, in an inductive manner. The main ideas of our construction can be used as well in other cases. A dual theory between lagrangians and hamiltonians (via Legendre transformations) is considered, in a similar way as R.Miron. A canonical way to induce a hamiltonian on an affine subbundle is given, too. AMS Subject Classification: 53B40, 53C60, 53C15.
A new category equivalence is proved in this paper, involving the two distinct categories of modu... more A new category equivalence is proved in this paper, involving the two distinct categories of modules, the covariant and the contravariant, respectively, released by Higgins and Mackenzie. The equivalence of the two categories is given when restricting to almost finitely generated projective modules and their allowed morphisms, defined in the paper. The equivalence is expressed by using generators. In particular, we obtain the well-known equivalence of the two categories of projective finitely generated modules; thus, our main result extends this classical one.
sont relativementcompactes alors on dit que F est de type fini compact. Aussi dans [8, Th´eor`eme ... more sont relativementcompactes alors on dit que F est de type fini compact. Aussi dans [8, Th´eor`eme 1.2.] prouve-t-onqu’un feuilletage de type fini compact est riemannien. Comme un feuilletage transversalementparall´elisable est riemannien, le r´esultat de Tarquini est am´elior´e par le r´esultat qui suit.Th´eor`eme 0.1 Un feuilletage F
Skew symmetric algebroids are generalizations of Lie algebroids, when the Jacobiator is not neces... more Skew symmetric algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural extension of the bundle and the anchor allows a Lie algebroid bracket. Some cohomologies and characteristic classes of a skew symmetric algebroid are constructed. We prove that characteristic classes of a skew symmetric algebroid are pull-backs of the characteristic classes of the base, as in the case of Lie algebroids.
The aim of the paper is to extend Lagrangian dynamics to Pfafi form dynamics, where a Pfafi form ... more The aim of the paper is to extend Lagrangian dynamics to Pfafi form dynamics, where a Pfafi form is a difierential form on a tangent bundle, non necessary closed. Considering the action of a Pfafi form on curves, given by a second order Lagrangian linear in accelerations, we obtain the equations of the flrst and second variations, using variational methods. In the non-singular case, considered mainly in the paper, the generalized Euler-Lagrange equation is a third order difierential equation. As examples, we flnd that the solutions of the difierential equations of motion of a charge in a fleld and the Euler equations for the rotational dynamics of a rigid body about its center of mass can be obtained as particular solutions of suitable Pfafi forms, with non-negative second variations.
We establish a one to one correspondence between the\linebreak connections $C^{(k-1)}$ (in the bu... more We establish a one to one correspondence between the\linebreak connections $C^{(k-1)}$ (in the bundle $% T^{k}M\rightarrow M$, used by R. Miron in his work on higher order spaces) and $C^{(0)}$ (in the affine bundle $% T^{k}M\rightarrow T^{k-1}M$, used for example in \cite{CSC1}).
The aim of the paper is to define Legendrians and their dual objects, Legendriens ⁄ , as generali... more The aim of the paper is to define Legendrians and their dual objects, Legendriens ⁄ , as generalizations of Lagrangians and ane Finsler and Lagrange spaces of higher order were studied in (9, 4) using the bundles of accelerations T k M ! M. Related to T k M, we consider in this paper the ane bundle T k M ! T ki1 M. A dual theory of higher order Hamilton spaces was recently studied in (5, 6) using the dual bundles T ki1 M £M T ⁄ M = T k⁄ M ! M. Related to T k M and T k⁄ M, we consider in this paper the ane exact Legendrian ⁄ of order k on M. The forms of closed Lagrangians and Lagrangians ⁄ of order k ‚ 1 are given by Propositions 1.1 and 1.5 respectively. The top components of a Legendrian or of a Legendrian ⁄ are particular cases of a top Legendrian and of a top Legendrian ⁄ respectively. If these are non-degenerated, the Legendrian, respectively the Legendrian ⁄ is called regular. We prove that the regular Legendrians and Legendrians ⁄ are in duality by Legendre transformations (T...
In this paper we continue our recent study of a manifold endowed with a singular or regular distr... more In this paper we continue our recent study of a manifold endowed with a singular or regular distribution, determined as the image of the tangent bundle under a smooth endomorphism, and generalize Bochner's technique to the case of a distribution with a statistical type structure. Following the theory of statistical structures on Riemannian manifolds and construction of an almost Lie algebroid on a vector bundle, we define the modified statistical connection and exterior derivative on tensors. Then we introduce the Weitzenbock type curvature operator on tensors and derive the Bochner-Weitzenbock type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian on a distribution.
In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic... more In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3, 4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.
The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dep... more 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 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.
International Journal of Theoretical Physics, 2007
Affine hamiltonians are defined in the paper and their study is based especially on the fact that... more Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine hamiltonians and lagrangians of order k ≥ 2 are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine familtonian h is equivalent with Euler-Lagrange equation of its dual lagrangian L. Zermelo condition is also studied and some non-trivial examples are given.
The aim of this paper is to extend from manifolds to vector bundles some classical geometric obje... more The aim of this paper is to extend from manifolds to vector bundles some classical geometric objects, associated with Lagrange and Hamilton metrics. Considering vector bundles endowed with almost Lie structures, defined in [24] by one of the authors, some geometric objects like R-(semi)sprays and R-connections of Cartan type are defined and studied. It is proved that the Lagrange equations deduced for Lie algebroids by A. Weinstein have a similar form for almost Lie structures.
The purpose of this paper is to prove that each of the following conditions is equivalent to that... more The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation F is riemannian: 1) the lifted foliation F^r on the r-transverse bundle ν ^r F is riemannian for an r≥ 1; 2) the foliation F_0^r on a slashed ν_∗^r F is riemannian and vertically exact for an r≥ 1; 3) there is a positively admissible transverse lagrangian on a ν_∗^r F, for an r≥ 1. Analogous results have been proved previously for normal jet vector bundles.
The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first... more The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. For instance, we obtain that the E-Chern form of E^1,0 associated to an almost complex connection ∇ on E can be expressed in terms of the matrix J_ER, where J_E is the almost complex structure of E and R is the curvature of ∇. Also, we consider a metric product connection associated to an almost Hermitian Lie algebroid and we prove that the mean curvature section of E^0,1 vanishes and the second fundamental 2--form section of E^0,1 vanishes iff the Lie algebroid is Hermitian.
The aim of the paper is to construct some Godbillon-Vey classes of a family of regular foliations... more The aim of the paper is to construct some Godbillon-Vey classes of a family of regular foliations, defined in the paper. These classes are cohomology classes on the manifold or on suitable open subsets. Some examples are also considered.
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Papers by Paul Popescu