Timelike $B_2$-slant helices in Minkowski space $E_1^4$

Ahmad T. Ali and Rafael López

Ahmad T. Ali Al-Azhar University, Faculty of Science Mathematics Department, Nasr City, 11448, Cairo, Egypt
Rafael López Universidad de Granada Departamento de Geometría y Topología 18071 Granada, Spain


Abstract: We consider a unit speed timelike curve $\alpha $ in Minkowski 4-space ${\mathbf{E}}_1^4$ and denote the Frenet frame of $\alpha $ by $\lbrace {\mathbf{T}}, {\mathbf{N}}, {\mathbf{B}}_1, {\mathbf{B}}_2\rbrace $. We say that $\alpha $ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of ${\mathbf{E}}_1^4$. In this work we study those helices where the function $\langle {\mathbf{B}}_2,U\rangle $ is constant and we give different characterizations of such curves.

AMSclassification: primary 53C50; secondary 53B30.

Keywords: Minkowski space, timelike curve, Frenet equations, slant helix.