TAILIEUCHUNG - On the geometry of null curves in the minkowski 4-space

In this paper, we study the basic results on the general study of null curves in the Minkowski 4-space R4 1. A transversal vector bundle of a null curve in R4 1 is constructed using a frenet Frame consisting of two real null and two space-like vectors. The null curves are characterized by using the Frenet frame. | Turk J Math 33 (2009) , 265 – 272. ¨ ITAK ˙ c TUB doi: On the geometry of null curves in the minkowski 4-space ˙ R. Aslaner, A. Ihsan Boran Abstract In this paper, we study the basic results on the general study of null curves in the Minkowski 4-space R41 . A transversal vector bundle of a null curve in R41 is constructed using a frenet Frame consisting of two real null and two space-like vectors. The null curves are characterized by using the Frenet frame. Key Words: Null curves, Minkowski space, Transversal vector bundle. 1. Introduction Definition The Minkowski 4 -space is the space R4 with the Lorentzian inner product g(x, y) = −x0 y0 + x1 y1 + x2 y2 + x3 y3 for all x, y ∈ R4 and will be denoted in the future by R41 . With respect to the standard basis of R41 , the matrix of g is η = diag(−1, 1, 1, 1). Definition A non-zero vector x of R41 is called space-like if g(x, x) > 0 , time-like if g(x, x) 0, 2 DT T is a space-like vector field, so we can take DT T = W1 which implies that h = 0 and k1 = 1 in the first equation of (). Thus h = 0 implies that t is the distinguished parameter for C and by Remark , C is a non-geodesic in R41 . By taking the derivative of W1 with respect to T , we have 1 DT W1 = √ (cosh t, sinh t, − cos t, sin t, ) 2 268 () ˙ ASLANER, IHSAN BORAN Choosing W2 = √1 (sinh t, cosh t, sin t, cos t), 2 and taking the derivative with respect to T , we have 1 DT W2 = √ (cosh t, sinh t, cos t, − sin t, ) = T. 2 This implies that k3 = −1, k4 = 0 in equation () and we obtain 1 N = √ (− cosh t, − sinh t, cos t, − sin t, ). 2 By taking the derivative of N with respect to T , we have 1 DT N = √ (− sinh t, − cosh t, cos t, − sin t, ) = −W2 . 2 This implies that k2 = 0 in equation (), so the harmonic curvatures H1 and H2 of C are indefinite. 3. The characterizations of null helices in minkowski 4-space R41 In the Euclidean space R3 , a helix satisfies that its tangent makes a constant angle with a fixed

• Toán học
• Null curves
• Minkowski space
• Transversal vector bundle
• Minkowski 4 space
• Distinguished Frenet frame
• Null curve
• Frenet frame
• Cartan curvature
• Curvature functions