Loxodromes on non-degenerate helicoidal surfaces in Minkowski space–time

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

“…Throughout the article, we are not interested in degenerate case. If W > 0, then the surface X in E 4 1 is spacelike and if W < 0, then it is a timelike surface. Let {e 1 , e 2 , N 1 , N 2 } be a local orthonormal frame field on X in E 4 1 such that e 1 , e 2 are tangent to X and N 1 , N 2 are normal to X.…”

“…If W > 0, then the surface X in E 4 1 is spacelike and if W < 0, then it is a timelike surface. Let {e 1 , e 2 , N 1 , N 2 } be a local orthonormal frame field on X in E 4 1 such that e 1 , e 2 are tangent to X and N 1 , N 2 are normal to X. Then, the coefficients of the second fundamental form of X according to N i , (i = 1, 2) are given by…”

“…If the mean curvature vector H of X is zero, then X is said to be a minimal (maximal) surface in E 4 1 and if the Gauss curvature of X is zero, then it is called as developable (flat) surface in E 4 1 . Also, X is said to be a marginally trapped surface if the mean curvature vector H is lightlike.…”

“…Let G(2, 4) denote the Grassmanian manifold consisting of all oriented 2−planes through the origin in E 4 1 and 2 E 4 1 the vector space obtained by the exterior product of 2−vectors in E 4 1 . Let f i 1 ∧ f i 2 and g i 1 ∧ g i 2 be two vectors in…”

“…In Minkowski 4-space, there are three types rotation with 2-dimensional axis, called elliptic, hyperbolic and parabolic rotations which leave the spacelike, timelike and degenerate planes invariant. By using these rotations and the definition of helicoidal surface, M. Babaarslan and N. Sönmez gave the parametrizations of three types of helicoidal surfaces in Minkowski 4-space, [4].…”