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On natural symmetries on slit tangent bundles of Finsler manifolds

Mohamed Tahar Kadaoui Abbassi, Abderrahim Mekrami

Abstract

In this paper, we introduce a broad class of metrics on the slit tangent bundle of Finsler manifolds, termed \emph{$F$-natural metrics}. These metrics parallel the well-established $g$-natural metrics on the tangent bundles of Riemannian manifolds and are constructed using six real functions defined over the domain of positive real numbers. We provide an in-depth characterization of conformal, homothetic, and Killing vector fields derived from specific lifts of vector fields and tensor sections on the slit tangent bundle, which is equipped with a general pseudo-Riemannian $F$-natural metric.

On natural symmetries on slit tangent bundles of Finsler manifolds

Abstract

In this paper, we introduce a broad class of metrics on the slit tangent bundle of Finsler manifolds, termed \emph{-natural metrics}. These metrics parallel the well-established -natural metrics on the tangent bundles of Riemannian manifolds and are constructed using six real functions defined over the domain of positive real numbers. We provide an in-depth characterization of conformal, homothetic, and Killing vector fields derived from specific lifts of vector fields and tensor sections on the slit tangent bundle, which is equipped with a general pseudo-Riemannian -natural metric.

Paper Structure

This paper contains 13 sections, 23 theorems, 136 equations.

Table of Contents

  1. Finsler geometry revisited
  2. $F$-natural metrics on slit tangent bundles of Finsler manifolds
  3. The Levi-Civita Connection of an $F$-natural metric
  4. Incompressibility of the Geodesic flow
  5. When are the fibers of the slit tangent bundle totally geodesic?
  6. Symmetries on slit tangent bundles of Finsler manifolds
  7. Horizontal lift vector fields
  8. Vertical lifts of vector fields
  9. Complete lifts of vector fields
  10. Vertical vector fields of the form $\iota P$
  11. The Liouville vector field
  12. The Geodesic vector field
  13. Proofs of technical results

Key Result

Lemma 1

For any vector fields $X, Y \in \mathfrak{X}(M)$, we have the following identities

Theorems & Definitions (34)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 24 more