An Advanced Approach to Bertrand Curves in 4-Dimensional Minkowski Space

This work presents a novel and comprehensive approach to the study of Bertrand curves in 4-dimensional Minkowski space (R 4 1 ). We introduce a new Frenet frame specifically tailored for analyzing Bertrand curves in the (1, 3)-normal plane, which allows us to derive significant relationships between the curvature functions κ1, κ2, and κ3. Our analysis provides new formulas and explicit conditions for these curvatures, offering a deeper understanding of their geometric properties in R 4 1 . We investigate four distinct cases of Bertrand curves, each characterized by specific conditions on the curvature functions. For each case, we derive explicit solutions and relationships, demonstrating the versatility of our approach. Furthermore, we establish the existence of a Bertrand mate curve ζ ∗ for a given Bertrand curve ζ and derive the parameter λ that defines the mate curve. This parameter is expressed in terms of the curvature functions, providing a clear connection between the original curve and its mate. To illustrate the practical application of our theoretical results, we provide detailed examples of Bertrand curve pairs in R 4 1 . These examples include the explicit construction of the Frenet frames and the computation of the associated curvature functions, showcasing the effectiveness of our methodology.