Golden Finsler Geometry: Local Properties and Global Deformations

Abstract

We introduce the concept of a golden Finsler structure on a finite-dimensional smooth manifold $M$ and investigate it from both local (coordinate-based) and global (coordinate-free) perspectives. Locally, we explicitly compute the fundamental metric tensor, establish the positive definiteness condition, and derive the geodesic spray coefficients. Furthermore, we investigate the projective flatness of the golden $(\alpha, \beta)$-metric and prove the non-existence of almost rational golden $(\alpha, \beta)$-metrics. Globally, we define the golden Finsler change $\widetilde{F}$ of a base Finsler metric $F$ and examine its geometric properties utilizing a special concurrent $\pi$-vector field. We explicitly determine how fundamental non-linear structures, including the Barthel and Berwald connections, transform under this change. Finally, we prove that $\widetilde{F}$ and $F$ cannot be projectively related.