On Concircular Transformations In Finsler Geometry

TL;DR

This paper investigates the properties of geodesic circles and concircular transformations in Finsler geometry, characterizing vector fields and curvature invariants through PDEs and connections.

Contribution

It introduces PDE characterizations of concircular vector fields and establishes conditions for conformal and concircular relationships in Finsler metrics.

Findings

Characterization of concircular vector fields via PDEs.

Necessary and sufficient conditions for conformal and concircular vector fields.

Invariant curvature properties under conformal and concircular transformations.

Abstract

A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we apply Lie derivatives and the Cartan $Y$-connection to study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on the tangent bundle, and then we obtain respective necessary and sufficient conditions for a concircular vector field to be conformal and a conformal vector field to be concircular. We also show conditions for two conformally related Finsler metrics to be concircular, and obtain some invariant curvature properties under conformal and concircular transformations.