Right-invariance and topological equivalence of the CIPP Finsler metric in the half-Lie group setting

Determine whether, for a right-invariant magnetic system (G, 𝒢, σ) on a half-Lie group with weakly exact σ and universal cover π: Ĝ → G with lifted metric Ĝ, the Finsler metric obtained via the Contreras–Iturriaga–Paternain–Paternain construction to conjugate the lifted magnetic flow above Mañé’s critical value is Ĝ-right-invariant and whether it induces on Ĝ the same topology as the lifted strong Riemannian metric Ĝ.

Background

In the finite-dimensional setting, for energies above Mañé’s critical value, the construction of Contreras–Iturriaga–Paternain–Paternain (CIPP) produces a Finsler metric whose geodesic flow coincides with the magnetic flow on the universal cover. The present paper extends the Hopf–Rinow theorem to right-invariant magnetic systems on half-Lie groups and, for energies above Mañé’s critical value, constructs an explicit Ĝ-right-invariant Randers Finsler metric 𝔽^κ on the universal cover Ĝ whose geodesic flow matches the lifted magnetic flow.

However, the authors note obstacles in directly adapting the CIPP construction to the infinite-dimensional half-Lie group setting: it is unclear whether the Finsler metric arising from the CIPP procedure is Ĝ-right-invariant and whether it induces the same topology as the lifted strong Riemannian metric Ĝ on Ĝ. These properties are essential for the Hopf–Rinow-type arguments developed in the paper, prompting the explicit open question stated below.