On the local structure of A-jet spaces 1

mathem.pub.ro

Within the framework of first order jet-Generalized Lagrange Spaces, the present survey article provides a survey presenting the work of a Romanian research group in the field of d−geometric structures on the first order jet space J 1 (T, M). Recent advances and actual open questions regarding these basic distinguished structures-which extend the corresponding generalized Lagrange, Lagrange, Finsler and Riemann d−structures of the tangent bundle framework, are described.

Revista Matemática Complutense, 1999

Wc introduce aud study the le-jet ampleneas aud the le-jet spannedness for a vector bundle, E, on a projective manifold. We obtain different characterizationa of projective apace in ternís of such positivity properties for E. Wc compare dic 1-jet ampleneas with different notions of very ampleneas in the literature.

Journal of Algebra

If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.

Journal of Pure and Applied Algebra, 2011

We collect a few results about jets of line bundles on curves and Wronskians, with a special emphasis to those arising from the canonical involution of a hyperelliptic curve.

2004

Jets of a manifold M can be described as ideals of C1(M). This way, all the usual processes on jets can be directly referred to that ring. By using this fact, we give a very simple construction of the contact system on jet spaces. The same way, we also dene the contact system for the recently considered A-jet spaces, where A is a Weil algebra. We will need to introduce the concept of derived algebra.

Czechoslovak Mathematical Journal

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2010

This is the first of two papers devoted to the proof of Zilber’s dichotomy for the case of difference-differential fields of characteristic zero. In this paper we use the techniques exposed in [9] to prove a weaker version of the dichotomy, more precisely, we prove the following: in DCFA the canonical base of a finite-dimensional type is internal to the fixed field of the field of constants. This will imply a weak version of Zilber’s dichotomy: a finite-dimensional type of SU -rank 1 is either 1-based or non-orthogonal to the fixed field of the field of constants. Resumen El presente es el primero de dos art́ıculos dedicados a la demostración de la dicotomı́a de Zilber para el caso de los campos difernciales de diferencia de caracteŕıstica cero. En éste art́ıculo utilizamos las técnicas desarrolladas en [9] para demostrar una versión débil de la dicotomı́a: un tipo de dimensión finita y de rango SU igual a 1 es modular o no ortogonal al campo fijo del campo de constantes.

1998

We characterize the triples (X,L,H), consisting of holomorphic line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J_k(L), is isomorphic to a direct sum H+...+H. Given the geometrical constrains imposed by a projectivized line bundle being a product of the base and a projective space it is natural to expect that this would happen only under very rare circumstances. It is shown, in fact, that X is either an Abelian variety or projective space. In the former case L\cong H is any line bundle of Chern class zero. In the later case for k a positive integer, L=O_{P^n}(q) with J_k(L)=H+...+H if and only if H=O_{P^n}(q-k) and either q\ge k or q\le -1.

2001

This paper is a continuation of [8], where we give a construction of the canonical Pfaff system Ω(M ` m) on the space of (m, `)-velocities of a smooth manifold M . Here we show that the characteristic system of Ω(M ` m) agrees with the Lie algebra of Aut( m), the structure group of the principal fibre bundle M̌ ` m −→ J ` m(M), hence it is projectable to an irreducible contact system on the space of (m,`)-jets (= `-th order contact elements of dimension m) of M . Furthermore, we translate to the language of Weil bundles the structure form of jet fibre bundles defined by Goldschmidt and Sternberg in [2]. 1. The characteristic system of Ω(M m) It is well known that Aut(Rm) is a Lie group whose Lie algebra is isomorphic to Der(Rm,Rm) (see [4, 5]); we are going to prove this result in a form which we will need later. The elements of Aut(Rm) are, in particular, linear automorphisms of R m; therefore if ξ̄ is the infinitesimal generator of a 1-parameter subgroup {τt} of Aut(Rm), we can as...

2000

Section 1 presents the main properties of the differentiable structure of the jet fibre bundle of order one J(T, M). Section 2 introduces an important collection of geometrical objects on J(T, M) as the d-tensors, the temporal and spatial sprays and the harmonic maps induced by these sprays. Moreover, we show that the notion of harmonic map induced by the sprays is a natural generalization of the classical notion of harmonic map between two Riemannian manifolds. In Section 3 we present the connection between the temporal and spatial sprays and the important notion of nonlinear connection on J(T, M). Section 4 studies the problem of prolongation of vector fields from T ×M to 1-jet space J(T, M). Mathematics Subject Classification: 53C07, 53C43, 53C99