The properly Helmholtz plane as Finsler geometry

G.G. Mikhailichenko has built the complete classification of two-dimensional phenomenologically symmetric geometries, i.e. geometries for which the six mutual distances between the four arbitrary points are functionally connected. In these geometries, the distance is understood in the generalized sense as the value of a function called a metric. The validity of metric axioms is not supposed. All these geometries are endowed with the maximum mobility, that is, there are groups of motions of maximum dimensionality equal to $3$. Classification of such two-dimensional geometries includes both well-known geometries (Euclidean, the pseudo-Euclidean, symplectic, spherical, etc.), and unknown ones (the Helmholtz, pseudo-Helmholtz, dual Helmholtz, and simplicial geometries).
In this paper, we use methods of Finsler geometry to study the properly Helmholtz twodimensional geometry. In the first section, we introduce the definition of the properly Helmholtz plane, and then we prove that it is a positive definite Finsler space (we check homogeneity and positivity of the metric function, as well as the positive definiteness of the Finsler metric tensor). The second section defines the properly Helmholtz two-dimensional manifold and proves that it is also a positive definite Finsler space. Then we calculate the basic Finsler tensor $C_{ijk}$ and additional $A_{ijk}$ tensor. With the help of these tensors, we find the Finsler scalar $J$ and prove that the special Finsler curvature tensor $S^i_{jkl}$ for the properly Helmholtz two-dimensional manifold is zero.