The pseudo-Helmholtz and dual Helmholtz planes with the Finsler geometry

There exists 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 a generalized sense as the value of a function called the metric function. Axioms of a metric are not obligatorily satisfied. For all these geometries, groups of motion are three-dimensional. The classification of such two-dimensional geometries includes both well-known geometries (Euclidean, pseudo-Euclidean, symplectic, spherical, etc.), and unknown ones (the properly Helmholtz, pseudo-Helmholtz, dual Helmholtz, and simplicial geometries).
In this paper, we use methods of Finsler geometry to study the pseudo-Helmholtz and dual Helmholtz two-dimensional phenomenologically symmetric geometries. In particular, in the first section, we introduce the definition of pseudo-Helmholtz and dual Helmholtz planes, and then prove that they are positive definite Finsler spaces (homogeneity and positivity of the metric function, as well as the positive definiteness of the Finsler metric tensor are verified), though, in contrast to the actual Helmholtz geometry, with some restrictions on the domain. In the second section, the pseudo-Helmholtz two-dimensional manifold is defined and it is proved that it is a positive definite Finsler space for $|\beta|>1$ in a certain domain. Then, the metric tensor $g_{ij}$, basic Finsler tensor $C_{ijk}$, and additional tensor $A_{ijk}$ are calculated. With these tensors, the Finsler scalar $\mathrm{J}$ is obtained and it is proved that the special Finsler curvature tensor $S^i_{jkl}$ for the two-dimensional pseudo-Helmholtz manifold is zero. In the third section, the dual Helmholtz two-dimensional manifold is defined and it is proved that it is a positive definite Finsler space in the domain of definition. Then, as in the second section, the metric tensor, basic Finsler tensor $C_{ijk}$, and additional $A_{ijk}$ tensor are calculated. Then, it is proved that $\mathrm{J}=2$ and the special Finsler curvature tensor $S^i_{jkl}=0$.