Time-like constant slope surfaces and space-like Bertrand curves in Minkowski 3-space

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space ${\mathbb{S}}_1^2$ and introducing space-like height function on the unit speed time-like curves on ${\mathbb{S}}_1^2$, the invariants of the unit speed time-like curves on ${\mathbb{S}}_1^2$ and geometric properties of de Sitter evolutes of the unit speed time-like curves on ${\mathbb{S}}_1^2$ are studied. A relation between space-like Bertrand curves and helices is obtained. De Sitter Darboux images of space-like Bertrand curves are equal to de Sitter evolutes. The relations between time-like constant slope surfaces lying in the space-like cone and space-like Bertrand curves in Minkowski 3-space ${\mathbb{R}}_1^3$ are obtained.