On generalized Berwald surfaces with locally symmetric fourth root metrics

Abstract

Let $m=2l$ be a positive natural number, $l=1, 2, \ldots. $ A Finslerian metric $F$ is called an $m$-th root metric if its $m$-th power $F^m$ is of class $C^{m}$ on the tangent manifold $TM$. Using some homogenity properties, the local expression of an $m$-th root metric is a polynomial of degree $m$ in the variables $y^1$, $\ldots$, $y^n$, where $\dim M=n$. $F$ is locally symmetric if each point has a coordinate neighbourhood such that $F^m$ is a symmetric polynomial of degree $m$ in the variables $y^1$, $\ldots$, $y^n$ of the induced coordinate system on the tangent manifold. Using the fundamental theorem of symmetric polynomials, the reduction of the number of the coefficients depending on the position makes the computational processes more effective and simple. In the paper we present some general observations about locally symmetric $m$-th root metrics. Especially, we are interested in generalized Berwald surfaces with locally symmetric fourth root metrics. The main result (Theorem 1) is their intrinsic characterization in terms of the basic notions of linear algebra. We present a one-parameter family of examples as well. The last section contains some computations in 3D. They are supported by the MAPLE mathematics softwer (LinearAlgebra).