Spinor Representations of Special Curve Couples of Framed Curves in 3D Lie Groups

In this study, we aim to bring together the spinors, a useful concept from mathematics to physics, and Bertrand and Mannheim curves of framed curves in 3D Lie groups, that are a special singular curve in 3D Lie groups. For this purpose, we investigate the spinor representations of Bertrand and Mannheim curves of framed curves in 3D Lie groups with a bi-invariant metric. Based on the natural structures of Bertrand and Mannheim curves of framed curves and Lie groups, this study is a comprehensive study that includes spinor representations of both regular and singular Bertrand and Mannheim curves in 3D Lie groups. It is a comprehensive generalization of all the studies done in the spinor representation of relevant concepts, as it includes studies on spinor representations of both regular and singular Bertrand and Mannheim curves in the literature. Then, we determine the special cases of spinor equations of these special curve couples of framed curves. Additionally, we construct some geometric interpretations and results with respect to them.