On Asanov's Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem

Finsleroid-Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G.S. Asanov in 2006. As a special case of the general construction Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. The so-called Asanov's unicorns (Landsberg spaces that are not of Berwald type) were startlers in Finsler geometry although the existence of regular (y-global) Landsberg metrics that are not of Berwald type is an open problem up to this day. Independently of Asanov's works, the Finsleroid-Finsler metrics appeared in 2003 related to the invariance problem of the mixed Berwald curvature under conformal changes. In the special case of vanishing curvature the problem is due to M. Matsumoto: are there conformally equivalent non-Riemannian Berwald manifolds? Although the condition of the conformal invariance of the mixed Berwald curvature turned out to be too strong to admit non-homothetic conformal changes, the proof reduced the problem to the investigation of the so-called Asanov-type Finslerian metric functions called Finsleroid-Finsler metrics by Asanov three years later in 2006. Using some weakening of the conformal invariance of the mixed Berwald curvature Finsleroid-Finsler metrics with closed Finsleroid axis 1-forms are characterized as the solutions of a conformal rigidity problem in this paper. We are looking for (non-Riemannian) Finsler metrics admitting a (non-homothetic) conformal change such that the mixed curvature tensor of the Berwald connection contracted by the derivatives of the logarithmic scale function is invariant. We prove that any solution must be locally conformal to a Finsleroid-Finsler metric with closed Finsleroid axis 1-form. Conversely, a Finsleroid Finsler metric with closed Finsleroid axis 1-form admits a local conformal change satisfying the rigidity condition.