On (m¯ ,m)-Conformal Mappings and Their Geodesic Extension

Many geometrical models have been created and applied in different subjects of sciences, such as physics, astronomy, biology,\ldots This paper presents a generalization of the recently defined $(\bar m,m)$-conformal mappings of Riemannian spaces. In this article, the affine connections of Riemannian spaces and symmetric affine connection spaces are combined. In this way, the characteristics of structures without external effects (Riemannian spaces) and with external effects (symmetric affine connection spaces) are geometrized. The main contributions of this research are: 1. Invariants of $(\bar m,m)$-conformal mappings of Riemannian spaces (review) and of geodesic mappings of symmetric affine connection spaces (review); 2. Actions and variational calculi with respect to some invariants obtained in 1; 3. Generalized Einstein's equations with respect to the analyzed transformations with clear reductions to standard ones.