On (m¯,m)-Conformal Mappings of Leaves

This study extends our previous investigations of $(\overline{m},m)$-conformal mappings. While the earlier studies were primarily theoretical and included only limited applications, the present study focuses on the applicability of these mappings to problems in mechanics. First, we derive the fundamental invariants associated with $(\overline{m},m)$-conformal mappings of two-dimensional manifolds. Based on these invariants, we formulate the corresponding Lagrangean and investigate its variational properties. By combining the resulting Euler–Lagrange equations with the transformation laws for the Christoffel symbols induced by transformations between manifolds, we derive the corresponding Einstein field equations within the proposed geometric framework. Subsequently, we analyze the transformation properties of the Einstein equations and investigate the influence of the relevant geometric areas on these transformations. Particular attention is given to the extent to which the resulting equations preserve the fundamental physical structure. Finally, the theoretical framework is illustrated through an application to the motion of a leaf. The corresponding velocity, acceleration, jerk, force, momentum, and mass are derived within the proposed model. This example serves to demonstrate that the fundamental laws of classical mechanics remain invariant under the considered transformations, thereby providing an illustrative validation of the physical consistency of the proposed geometric approach.