2015/09/17 by Tanju Kahraman, Kahraman, Tanju, Mehmet Önder +1
Engineering · Mathematics · #53A04 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1509.05170
openalex publication_date 2015/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Mannheim curves are defined for immersed curves in 3-dimensional sphere S3 . The definition is given by considering the geodesics of S3. First, two special geodesics, called principal normal geodesic and binormal geodesic, of S3 are defined by using Frenet vectors of a curve immersed in S3. Later, the curve alpha is called a Mannheim curve if there exits another curve beta in S3 such that the principal normal geodesics of beta coincide with the binormal geodesics of S3 . It is obtained that if alpha and beta form a Mannheim pair then there exist a constant lambda that is not equal 0 and a non-constant function Mu such that Lambda.(kappaalpha)+M(Taualpha)=1 where kappaalpha, Taualpha are the curvatures of alpha. Moreover, the relation between a Mannheim curve immersed in S3 and a generalized Mannheim curve in E4 is obtained and a table containing comparison of Bertrand and Mannheim curves in S3 is introduced.