New characterizations of vector fields on Weil bundles

Journal of Mathematics, 2023

Let M be a smooth manifold and A a Weil algebra. We discuss the diferential forms in the Weil bundles (M A , π, M), and we established a link between diferential forms in M A and M as well as their cohomology. We also discuss the cohomology in.

Symmetry, Integrability and Geometry: Methods and Applications, 2012

We prove that a vector bundle π : E → M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole fibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229-239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1.

The purpose of this note on Poisson cohomology is to show that, if M is a Poisson manifold and if A is a Weil algebra, then the Weil bundle M A is a Poisson manifold. Therefore, we establish a relationship between Poisson cohomology H • {.,.}A (M A , A) with values in A and Poisson cohomology H • {.,.},R (M A ) with real values.

Annales Polonici Mathematici, 2019

The purpose of this note on Poisson cohomology is to show that, if M is a Poisson manifold and if A is a Weil algebra, then the Weil bundle M A is a Poisson manifold. Therefore, we establish a relationship between Poisson cohomology H • {.,.}A (M A , A) with values in A and Poisson cohomology H • {.,.},R (M A) with real values.

2016

Let M be a paracompact smooth manifold, A be a Weil algebra and M A be the associated Weil bundle. If is a linear connection on M , we give equivalent definition and the properties of the prolongation A to M A equivalent to the prolongation defined by Morimoto. When (M, g) is a pseudo-riemannian manifold, we show that the symmetric tensor g A of type (0, 2) defined by Okassa is nondegenerated. At the end, we show that, if is a Levi-Civita connection on (M, g), then A is torsion-free and g A is parallel with respect to A .

2015

Let M be a paracompact smooth manifold of dimension n; A a Weil algebra and M the associated Weil bundle. In this paper, we de ne the Schouten-Nijenhuis bracket on the C1(MA; A)-module X (M) of multivector elds on M considered as multi-derivations from C1(M) into C1(MA; A) and we show that the exterior algebra X (M) of multivector elds on M is a Lie graded algebra over A. To nish, we estabish an isomorphism between X(M) and the C1(MA; A)-module Lqalt( (M; A); C1(MA; A)) of skew-symmetric multilinear forms of degree q onto the C1(MA; A)-module (M; A) of di¤erential A-forms on M. Mathematics Subject Classi cation: 58A20, 58A32. Key words: Weil bundle, Weil algebra, Multivector elds, Schouten-Nijenhuis bracket.

2015

In this paper, we denote by A a Weil algebra, M a smooth manifold and M^{A} the associated Weil bundle and we study the properties of differential operators on M^{A} and construct the canonical 1-form when M^{A} is provided with a structure of Lie-Rinehart algebra.

International Mathematical Forum, 2014

Let M be a paracompact smooth manifold of dimension n, A a Weil algebra and M A the Weil bundle associated. We define and describe the notion of d-Poisson cohomology and of d A-Poisson cohomology on M A .

New Trends in Mathematical Science, 2019

Let (M, g)be a Riemannian manifold and T (M) its tangent bundle with the horizontal lift H ∇ of the affine connection ∇ of M. The aims of the present paper are to study conditions of infinitesimal affine transformation for vertical and horizontal vector fields in the tangent bundle and to give necessary conditions for vertical, complete and horizontal vector fields in the tangent bundle to be a harmonic vector fields with respect to the horizontal lift H ∇ of the affine connection ∇ of M.