Jet Berwald-Riemann-Lagrange Geometrization for Affine Maps between Finsler Manifolds

TL;DR

This paper introduces a new geometric framework for affine maps between Finsler manifolds, utilizing Berwald-Riemann-Lagrange geometry on jet spaces to characterize these maps through connections, torsions, and curvatures.

Contribution

It develops a natural Berwald-Riemann-Lagrange geometric structure on jet spaces to analyze affine maps between Finsler manifolds, providing new geometric insights.

Findings

Defined affine maps between Finsler manifolds.

Constructed Berwald-Riemann-Lagrange geometry on jet spaces.

Characterized affine maps via connections, torsions, and curvatures.

Abstract

In this paper we introduce a natural definition for the affine maps between two Finsler manifolds $(M, F)$ and $(N,\tilde F)$ and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;N)$, in the sense of a Berwald nonlinear connection $\Gamma^b_jet$, a Berwald $\Gamma^b_jet$-linear d-connection $B\Gamma^b_jet$, together with its d-torsions and d-curvatures, which geometrically characterizes the initial affine maps between Finsler manifolds.