On (m¯, m)-Conformal Mappings

Abstract

Conformal mappings between Riemannian spaces ¯RN and RN are defined by the explicit transformation of the metric tensor of the space ¯RN to the metric tensor of the space RN . Geodesic mapping between these two Riemannian spaces is a transformation that transforms any geodesic line of the space ¯RN to a geodesic line of the space RN . In this research, we defined an m-conformal line of a Riemannian space, which is geodesic if m = 0. Based on this definition, we involved the concept of ( ¯m, m)-conformal mapping as a transformation ¯RN → RN in which any ¯m-conformal line of the space ¯RN transforms to an m-conformal line of the space RN . The result of this research is the establishment of three invariants for these mappings. At the end of this research, we gave an example of a scalar geometrical object which may be used in physics.