On the Laurent Series Ring as a Loop Integral Constructed from Curvature Tensor Residues and Scalar Curvature Winding Numbers

This paper establishes a profound connection between the Laurent Series Ring in complex analysis and the curvature tensors in differential geometry. By interpreting curvature tensors as residues and scalar curvature as winding numbers within loop integrals, we demonstrate that the Laurent Series Ring can be effectively constructed through these geometric constructs. This synthesis bridges complex analysis and differential geometry, offering new insights into the interplay between analytic and geometric structures.