An observation on Asanov's Unicorn metrics

TL;DR

This paper investigates Asanov's Unicorn metrics within Finsler geometry, showing they are generalized Berwald spaces when the Finsleroid charge is constant, and characterizing Landsberg spaces in this context.

Contribution

It proves that Asanov's Unicorns are generalized Berwald manifolds if and only if the Finsleroid charge is constant, clarifying their geometric structure.

Findings

Unicorns belong to generalized Berwald manifolds when charge is constant.

Finsleroid-Finsler spaces are Landsberg if and only if they are generalized Berwald with semi-symmetric connection.

Characterization of Unicorns within the broader class of Finsler spaces.

Abstract

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanov's Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generalized Berwald space if and only if the Finsleroid charge is constant. Especially a Finsleroid-Finsler space is a Landsberg space if and only if it is a generalized Berwald manifold with a semi-symmetric compatible linear connection.