Marcela Popescu

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 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.

Curves with Special Aesthetics Generated by an Original Mechanism

Our goal is to create artificial beauty by generating new plane figures—curves not known in Geometry. The paper deals with an original mechanism that generates curves featuring special aesthetic properties. The mechanism synthesis was achieved starting from relations known from Trigonometry for multiple arcs. The angles considered for the aforementioned arcs are represented by the angles of the leading elements, correlated through a coefficient n. The mechanism has two leading elements and two dyads of type RPP, rotation-translation-translation. In the extended paper the cinematic schema of the mechanism is provided. Its structure is analyzed, the associated equations are introduced along with the obtained plots (many curves with special aesthetic properties). Comments are made with respect to the influence of the coefficient n used to correlate the angles of the leading elements over the shapes of the generated curves and over the number of close contours. The obtained curves are analyzed for both possible distinct cases: when both leading elements are rotating clockwise, respectively in opposite senses. The continuity of these curves is studied as well, close or open curves being obtained. Some mathematical properties of the curves generated by the above mechanism are studied (the curvature and the curvature radius in a current point of the curve, the equation of the tangent and the equation of the normal to the curve in a current point and the symmetries of the curve with respect to the coordinate axes (Ox axis and Oy axis) and with respect to the origin O).

EXTENDED AFFINE CLASSES OF LAGRANGIANS AND HAMILTONIANS RELATED TO CLASSICAL FIELD THEORIES

The aim of the paper is to establish a natural affine frame for affine Lagrangians and Hamiltonians, generalizing the well-known classical field theory. Scalar and volume-valued Lagrangians and Hamiltonians can be lifted to the new classes. Using the Hamilton–Jacobi principle, we analyze variational problems corresponding to actions defined by the affine Lagrangians and Hamiltonians. The extremals verify generalizations of the Euler–Lagrange and De Donder–Weyl PDEs. They improve the information about the dynamical solutions of the classical variational problems and refresh the Lagrange–Hamilton theories.

Induced Semi-Sprays and Connections on Submanifolds

Some kinds of induced semisprays and corresponding non-linear connections on submanifolds are studied in this paper. Some new definitions, as the vertical tangent bundle of a submanifold, are made. As an application, the semi-spray which corresponds to the Cartan non-linear connection of a Lagrange or a Finsler space induces the same objects on a submanifold.