Adiabatic limit and connections in Finsler Geometry

TL;DR

This paper explores the relationships between different connections in Finsler geometry, showing how the Cartan connection arises from the Bott connection via symmetrization and adiabatic limits, and discusses associated Chern-Simons forms.

Contribution

It establishes a novel link between Bott, Chern, and Cartan connections in Finsler geometry using adiabatic limit techniques and introduces a conformally invariant Chern-Simons type form.

Findings

Bott connection corresponds to the Chern connection on the projective sphere bundle.

Symmetrization of the Bott connection yields the Cartan connection.

A Chern-Simons type form with conformal properties is constructed.

Abstract

In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{LiuZ}, the Cartan connection can also be obtained through an adiabatic limit process. Furthermore, a Chern-Simons type form is defined and its conformal properties are discussed.