Natural connections for semi-Hamiltonian systems: The case of the $\epsilon$-system

TL;DR

This paper constructs a compatible connection on an $F$-manifold for semi-Hamiltonian systems, exemplified by the $ ext{epsilon}$-system, revealing flatness and commuting flows.

Contribution

It introduces a method to associate a flat, compatible connection to semi-Hamiltonian systems, specifically applied to the $ ext{epsilon}$-system, enhancing understanding of their geometric structure.

Findings

The constructed connection is flat.

Flat coordinates generate commuting flows.

Application to the $ ext{epsilon}$-system demonstrates the method.

Abstract

Given a semi-Hamiltonian system, we construct an $F$-manifold with a connection satisfying a suitable compatibility condition with the product. We exemplify this procedure in the case of the so-called $\epsilon$-system. The corresponding connection turns out to be flat, and the flat coordinates give rise to additional chains of commuting flows