On generalized Mannheim curves in Euclidean 4-space
Two curves \(C\), \(\hat{C}\) in Euclidean three-space are called ``Mannheim mates'' if there exists a bijection between their points such that the principal normal of \(C\) is the binormal at the corresponding point of \(\hat{C}\). The authors generalize this concept to curves in Euclidean four-space in the sense that the first axis of the Frenet frame of \(C\) is required to be contained in the plane spanned by the second and third axis of the Frenet frame of \(\hat{C}\). The authors prove that the first curvature \(k_1\) and the second curvature \(k_2\) of \(C\) necessarily satisfy a relation of the form \(k_1 = \alpha(k_1^2 + k_2^2)\) with a positive constant \(\alpha\), thus generalizing a well-known result on Mannheim curves in three-space. Under certain additional assumptions the converse is also shown to be true. Finally, the authors provide an explicit parametrization of a family of Mannheim curves in four-space that depends on a constant \(\alpha\) and two arbitrary smooth functions.
scientific article; zbMATH DE number 5640682