arXiv subjects

Willy Sarlet

Publications and source records attributed to Willy Sarlet.

4 recordsLinked to original sources

arXiv2013

Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over R

We study natural lifting operations from a bundle E over R to the dual bundle of its first-jet bundle. The main purpose is to define a complete lift of a type (1,1) tensor field on E and to understand all features of its construction. Various other lifting operations of tensorial objects on E are needed for that purpose. We prove that the complete lift of a type (1,1) tensor with vanishing Nijenhuis torsion gives rise to a Poisson-Nijenhuis structure on the dual of the first-jet bundle, and discuss in detail how the construction of associated Darboux-Nijenhuis coordinates can be carried out.

math.DG↗

arXiv2007

A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold

We review properties of so-called special conformal Killing tensors on a Riemannian manifold $(Q,g)$ and the way they give rise to a Poisson-Nijenhuis structure on the tangent bundle $TQ$. We then address the question of generalizing this concept to a Finsler space, where the metric tensor field comes from a regular Lagrangian function $E$, homogeneous of degree two in the fibre coordinates on $TQ$. It is shown that when a symmetric type (1,1) tensor field $K$ along the tangent bundle projection $τ: TQ\to Q$ satisfies a differential condition which is similar to the defining relation of special conformal Killing tensors, there exists a direct recursive scheme again for first integrals of the geodesic spray. Involutivity of such integrals, unfortunately, remains an open problem.

math.DG↗

arXiv2003

The Berwald-type linearisation of generalised connections

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.

math.DG↗

arXiv2002

Lie algebroid structures and Lagrangian systems on affine bundles

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.

math.DG↗