Ew forms of slant helices were presented in Minkowski space-time [6] and four-dimensional Euclidian spaces [7]. Within this paper, as provided in the Euclidean 4-space, we construct k-type helices and (k, m)kind slant helices according to the extended Darboux frame field EDFFK and EDFSK in four-dimensional Minkowski space E4 .Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Copyright: 2021 by the author. Licensee MDPI, Basel, Switzerland. This short article is definitely an open access short article distributed beneath the terms and situations on the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ four.0/).Symmetry 2021, 13, 2185. https://doi.org/10.3390/symhttps://www.mdpi.com/journal/symmetrySymmetry 2021, 13,2 of2. Geometric Preliminaries Minkowski space-time E4 may be the genuine (Z)-Semaxanib Purity & Documentation vector space R4 offered together with the indefinite flat 1 metric provided by , = -da2 da2 da2 da2 , two 3 1 4 where ( a1 , a2 , a3 , a4 ) is a rectangular coordinate program of E4 . We get in touch with E4 , , a Minkowski 1 4-space and denote it by E4 . We say that a vector a in E4 \0 can be a spacelike vector, a 1 1 lightlike vector, or possibly a timelike vector if a, a is constructive, zero, or negative, respectively. In certain, the vector a = 0 is usually a spacelike vector. The norm of a vector a E4 is defined by 1 a = | a, a |, plus a vector a satisfying a, a = 1 is known as a unit vector. For any two vectors a; b in E4 , if a, b = 0, then the vectors a and b are mentioned to be orthogonal vectors. 1 Let : I R E4 be an arbitrary curve in E4 ; if all of the velocity vectors of are 1 1 spacelike, timelike, and null or lightlike vectors, the curve is named a spacelike, a timelike, or even a null or lightlike curve, respectively [1]. A hypersurface inside the Minkowski 4-space is known as a spacelike hypersurface when the induced metric on the hypersurface can be a constructive definite Riemannian metric, and also a Lorentzian metric induced on the hypersurface is called a timelike hypersurface. The regular vector of your spacelike hypersurface is actually a timelike vector and the normal vector from the timelike hypersurface can be a spacelike vector. Let a = ( a1 , a2 , a3 , a4 ), b = (b1 , b2 , b3 , b4 ), c = (c1 , c2 , c3 , c4 ) R4 ; the vector solution of a, b, and c is defined using the determinant- e1 a1 abc = – b1 ce2 a2 b2 ce3 a3 b3 ce4 a4 , b4 cwhere e1 , e2 , e3 , and e4 are mutually orthogonal vectors (standard basis of R4 ) satisfying the equations [1]: e2 e3 e4 = e1 , e3 e4 e1 = e2 , e4 e1 e2 = – e3 , e1 e2 e3 = e4 .Let M be an oriented non-null hypersurface in E4 and let be a non-null normal 1 Frenet curve with speed v = on M. Let t, n, b1 , b2 be the moving Frenet frame along the curve . Then, the Frenet formulas of are: t = n vk1 n, n = – t vk1 t b1 vk2 b1 , b1 = – n vk2 n – t n b1 vk3 b2 , b2 = – b1 vk3 b1 exactly where t = t, t , n = n, n , b1 = b1 , b1 , and b2 = b2 , b2 , whereby t , n , b1 , b2 -1, 1 and t n b1 b2 = -1. The vectors , , , and (4) of a non-null frequent curve are given by = vt, = v t n v2 k1 n, = v – t n v3 k2 t n 3vv k1 v2 k1 n n b1 v3 k1 k2 b1 , 1 (4) = (. . .)t (. . .)n (. . .)b1 – t v4 k1 k2 k3 b2 .Symmetry 2021, 13,three Pinacidil References ofThen, for the Frenet vectors t, n, b1 , b2 and also the curvatures k1 , k2 , k3 of , we’ve got, n = b1 b2 , b1 b2 b1 = – n b2 , b2 = b1 , b2 b ,(4) b1 , n, k1 = , k3 = – t b2 2 4 2 , k2 = n three k1 k1 kt=Since the curve lies on M, if we denote the unit norma.
