Riemannian Geometry in Multiplicative Analysis: Curvature, Connections, and Isomorphic Structures

The goal of this article is to examine Riemann manifolds with the help of multiplicative arguments. Using the unique metric of multiplicative analysis (proportional metric), Christoffel symbols, connections and curvature tensor fields are designed on the manifolds built with this metric. Through this approach, we have scrutinized the multiplicative Riemann curvature of the multiplicative Euclidean space, uncovering its divergent structural attributes compared to the conventional Euclidean space. Moreover, with the help of the multiplicative Riemann curvature tensor, it has been revealed that the multiplicative Euclidean space is isomorphic to the traditional Euclidean space. Thus, the solutions of some problems that cannot be solved in the traditional Euclidean space will be obtained in the multiplicative space and with the help of isomorphic transformation, their solutions will be obtained in the Euclidean space. Also, illustrative examples are incorporated to facilitate a deeper comprehension of the discussed concepts.