On a new class of Finsler metrics

TL;DR

This paper introduces a new class of Finsler metrics called general (alpha,beta)-metrics, expanding the geometric understanding and generalization of existing (alpha,beta)-metrics with potential implications for Finsler geometry research.

Contribution

The paper defines and explores general (alpha,beta)-metrics, a novel class that broadens the scope of Finsler metrics beyond traditional (alpha,beta)-metrics and includes metrics structured by R. Bryant.

Findings

Introduces the concept of general (alpha,beta)-metrics.

Shows these metrics generalize classical (alpha,beta)-metrics.

Proposes non-classical methods related to beta-deformations.

Abstract

In this paper, the geometric meaning of (alpha,beta)-norms is made clear. On this basis, we introduce a new class of Finsler metrics called general (alpha,beta)-metrics, which are defined by a Riemannian metric and an 1-form. These metrics not only generalize original (alpha,beta)-metrics naturally, but also include some metrics structured by R. Bryant. The notion of general (alpha,beta)-metrics is one of the original ideas belongs to the first author(another one is beta-deformations intruduced in the paper "Deformations and Hilbert's Fourth Problem"). We believe that the researches on general (alpha,beta)-metrics will enrich Finsler geometry and the approaches offer references for further study. But it seems that the classical methods suitable for (alpha,beta)-metrics may not be suitable for them, the idea used in this paper, which is closely related to beta-deformations, is non-classical. Any communication or suggestion is welcome.