Malaya J. Mat. 4(3)(2016) 338–348 The Natural Lift of the Fixed Centrode of a Non-null Curve in Minkowski 3-Space Mustafa C ¸ alıs ¸kan a and Evren Erg ¨ un b, ∗ a Faculty of Sciences, Department of Mathematics, Gazi University, Ankara, Turkey. b C ¸ ar¸ samba Chamber of Commerce Vocational School, Ondokuz Mayıs University, Samsun, Turkey. Abstract In this study, we dealt with the natural lift curves of the fixed centrode of a non-null curve.Furthermore, some interesting result about the original curve were obtained, depending on the assumption that the natural lift curves should be the integral curve of the geodesic spray on the tangent bundle T ( S 2 1 ) and T ( H 2 0 ) . Keywords: Natural lift, geodesic spray, Darboux vector. 2010 MSC: 51B20, 53B30, 53C50. c 2012 MJM. All rights reserved. 1 Introduction Thorpe gave the concepts of the natural lift curve and geodesic spray in [12]. Thorpe provied the natural lift α of the curve α is an integral curve of the geodesic spray iff α is an geodesic on M.C ¸ alıs ¸kan at al. studied the natural lift curves of the spherical indicatries of tangent, principal normal, binormal vectors and fixed centrode of a curve in [11]. They gave some interesting results about the original curve, depending on the assumption that the natural lift curve should be the integral curve of the geodesic spray on the tangent bundle T ( S 2 ) . Some properties of M-vector field Z defined on a hypersurface M of M were studied by Agashe in [1]. M-integral curve of Z and M-geodesic spray are defined by C ¸ alıs ¸kan and Sivrida˘ g. They gave the main theorem: The natural lift α of the curve α (in M) is an M-integral curve of the geodesic spray Z iff α is an M-geodesic in [5]. Bilici et al. have proposed the natural lift curves and the geodesic sprays for the spherical indicatrices of the the involute evolute curve couple in Euclidean 3-space. They gave some interesting results about the evolute curve, depending on the assumption that the natural lift curve of the spherical indicatrices of the involute should be the integral curve on the tangent bundle T ( S 2 ) in [3]. Then Bilici applied this problem to involutes of a timelike curve in Minkowski 3-space (see [4]). Erg ¨ un and C ¸ alıs ¸kan defined the concepts of the natural lift curve and geodesic spray in Minkowski 3-space in [7]. The anologue of the theorem of Thorpe was given in Minkowski 3-space by Erg ¨ un and C ¸ alıs ¸kan in [7]. C ¸ alıs ¸kan and Erg ¨ un defined M-vector field Z, M-geodesic spray, M-integral curve of Z, M-geodesic in [6].The anologue of the theorem of Sivrida ˘ g and C ¸ alıs ¸kan was given in Minkowski 3-space by Erg ¨ un and C ¸ alıs ¸kan in [5]. Walrave characterized the curve with constant curvature in Minkowski 3-space in [12]. In differential geometry, especially the theory of space curve, the Darboux vector is the areal velocity vector of the Frenet frame of a spacere curve. It is named after Gaston Darboux who discovered it. In term of the Frenet-Serret apparatus, the darboux vector W can be expressed as W = τ T + κ B, details are given in Lambert et al. in [8]. In this study,we studied the fixed centrode curve of a curve and characterized the curve if the natural lift of the fixed centrode curve is an integral curve of the geodesic sprays. ∗ Corresponding author. E-mail address: mustafacaliskan@gazi.edu.tr (Mustafa C ¸ alıs ¸kan), eergun@omu.edu.tr (Evren Erg ¨ un).