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

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 com...

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Bibliographic Details
Main Author: Willy Sarlet
Format: Article
Language:English
Published: National Academy of Science of Ukraine 2007-02-01
Series:Symmetry, Integrability and Geometry: Methods and Applications
Subjects:
special conformal Killing tensors
Finsler spaces
Online Access:http://www.emis.de/journals/SIGMA/2007/024/
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spelling doaj-85406a9d81fa45eaa70c2ab839eca0b32020-11-25T01:25:41ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592007-02-013024A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler ManifoldWilly SarletWe 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 $au: TQ ightarrow Q$ 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.http://www.emis.de/journals/SIGMA/2007/024/special conformal Killing tensorsFinsler spaces
collection DOAJ
language English
format Article
sources DOAJ
author Willy Sarlet
spellingShingle Willy Sarlet
A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
Symmetry, Integrability and Geometry: Methods and Applications
special conformal Killing tensors
Finsler spaces
author_facet Willy Sarlet
author_sort Willy Sarlet
title A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
title_short A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
title_full A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
title_fullStr A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
title_full_unstemmed A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
title_sort recursive scheme of first integrals of the geodesic flow of a finsler manifold
publisher National Academy of Science of Ukraine
series Symmetry, Integrability and Geometry: Methods and Applications
issn 1815-0659
publishDate 2007-02-01
description 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 $au: TQ ightarrow Q$ 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.
topic special conformal Killing tensors
Finsler spaces
url http://www.emis.de/journals/SIGMA/2007/024/
work_keys_str_mv AT willysarlet arecursiveschemeoffirstintegralsofthegeodesicflowofafinslermanifold
AT willysarlet recursiveschemeoffirstintegralsofthegeodesicflowofafinslermanifold
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