Differential Equations of the Space-Like Loxodromes on the Helicoidal Surfaces in Minkowski 3-Space

Time-like loxodromes on helicoidal surfaces in Minkowski 3-space

“…Also they gave some important applications of them. Differential equations of the space-like loxodromes on helicoidal surfaces in Minkowski 3-space were found by Babaarslan and Kayacik [4].…”

“…where f (u) (u) and λ ∈ R + . If we take λ = 0 in the equations (3)-(5), then we have the rotational surfaces in E 3 1 (see [4], [5]). A basis of the tangent plane at each point of helicoidal surface H can be given by {H u , H v }.…”

Bour's Theorem of Spacelike Surfaces in Minkowski 4--Space

“…Many mathematicians have obtained significant results regarding helicoidal surfaces in different ambient spaces (see [12,5,7,22,10,23,11,24,26,27,1,2,3]). One of the most notable result is a generalization of the relationship between a right helicoid and a catenoid.…”

“…After these works, it seems natural question whether Bour's theorem have a counterpart in Minkowski space. For this purpose, T. Ikawa [21] studied Bour's theorem of spacelike and timelike helicoidal surfaces in Minkowski 3-space with non-null and null axis and he determined the parametrizations the pair of isometric surfaces which have same Gauss map in E 3 1 . Then, E. Güler and A. T. Vanlı examined Bour's theorem on generalized helicoid with null axis in Minkowski 3-space in [18].…”

“…The notion of Minkowski 4-space or Minkowski space-time is strongly related to Einstein's special theory of relativity and and it was proposed as a model for spacetime in the special theory of relativity by Minkowski in 1907, (see [29]). The Minkowski 4-space has more complicated and richer geometric structures than the Euclidean 4-space due to the difference between their metrics, angles and the motions of particles (see [3]).…”