Characterization of Bundles of Lines and Their Properties in the Euclidean 3-Space

In this paper, we study and define the notion of bundles of lines in the 3-dimensional Euclidean space, we also investigate their essential geometric aspects. Starting from a regular surface, we construct an orthonormal moving frame on the initial surface and employ it to describe the local behavior of associated line bundles also, and derive the fundamental equations of the bundles of lines. Then, we utilize the tools of differential forms to prove some basic theorems relevant to this study. Using the formalism of differential forms and the basic theorems, several foundational theorems related to these bundles are established, providing a unified framework for studying families of lines generated by surfaces which open new perspectives for applications in classical differential geometry and providing also, a geometric framework for further analysis and applications.