Distinguished Torsion, Curvature and Deflection Tensors in the Multi-Time Hamilton Geometry

TL;DR

This paper develops the geometric framework of multi-time Hamilton geometry on the dual 1-jet bundle, introducing key tensors like torsion, curvature, and deflection to characterize the structure.

Contribution

It introduces the concept of nonlinear connections and describes the torsion, curvature, and deflection tensors in the context of multi-time Hamilton geometry.

Findings

Defined the nonlinear connection on the dual 1-jet bundle.

Derived the adapted components of torsion, curvature, and deflection tensors.

Provided a geometric characterization of the multi-time Hamilton phase space.

Abstract

The aim of this paper is to present the main geometrical objects on the dual 1-jet bundle $J^{1*}(\cal{T},M)$ (this is the polymomentum phase space of the De Donder-Weyl covariant Hamiltonian formulation of field theory) that characterize our approach of multi-time Hamilton geometry. In this direction, we firstly introduce the geometrical concept of a nonlinear connection $N$ on the dual 1-jet space $J^{1*}(\cal{T},M)$. Then, starting with a given $N$-linear connection $D$ on $J^{1*}(\cal{T},M)$, we describe the adapted components of the torsion, curvature and deflection distinguished tensors attached to the $N$-linear connection $D$.