Time-Like Constant Slope Surfaces and Space-Like Bertrand Curves in Minkowski 3-Space

Abstract

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space S 2 1 and introducing space-like height function on the unit speed time-like curves on S 2 1 ; the invariants of the unit speed time-like curves on S 2 1 ;and geometric properties of de Sitter evolutes of the unit speed time-like curves on S 2 1 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 R 3 1 are obtained.

Some Results on p-Shape Curvatures of Non-Lightlike Space Curves

“…A pseudo-similarity (p-similarity) of E 3 1 is a composition of a homothety (dilatation), a pseudo-orthogonal map and a translation. Then, any p-similarity f : E 3 1 → E 3 1 can be expressed by…”

“…We give some information regarding the similarity geometry of non-null curves in E 3 1 . In the next section, we examine a relation between a non-lightlike space curve α and its pseudospherical tangential indicatrix c and we find that the p-shape torsion of α is equal to the geodesic curvature of c. Furthermore, we show that the pseudo-spherical Darboux images of α coincide with the pseudo-spherical evolutes of c. We represent the notion of similar helix and give a characterization of the similar helix in E 3 1 . Lastly, using the tangent indicatrix, we obtain the parametrizations of all non-lightlike self-similar curves, whose p-shapes are real constants.…”

On the evolute offsets of ruled surfaces in Minkowski 3-space

“…If γ(s) is a space-like curve in L 3 , the Frenet formulae of γ(s) are given by ( [16]) If γ(s) is a time-like curve in L 3 , the Frenet formulae of γ(s) are given by ( [16]) It is called the pseudospherical Frenet frame of the pseudospherical curve γ(s). If γ is a space-like curve, then the vector g is time-like when γ is on S 2 1 , and the vector g is space-like when γ is on H 2 . Similarly, if the curve γ is time-like, then the vector g is space-like.…”

“…Here γ is on H 2 when ϵ = 1 , and γ is on S 2 1 when ϵ = −1. (2) If γ is a pseudospherical time-like curve,…”

“…The geometry of curves and surfaces in Euclidean 3-space E 3 represented for many years a popular topic in the field of classical differential geometry. Increasing interest in the theory of curves has led to the development of special curves to be examined.…”