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In this paper, the Mannheim mate curves of the proper biharmonic curves in Cartan-Vranceanu 3-dimensional spaces (M, ds 2 l,m), with l 2 4m and m 0 are studied. We give the definition of the Mannheim mate of a proper biharmonic curve and give the explicit parametric equations of that Mannheim mate curve in Cartan-Vranceanu 3-dimensional space. Moreover, we show that the distance between corresponding points of the Mannheim pairs is constant in Cartan-Vranceanu 3-dimensional spaces.

Non-geodesic biharmonic curves in Cartan-Vranceanu 3-dimensional spaces are studied in [3]. In this paper, we characterize parametric equations of Bertrand mate of a biharmonic curve in the Cartan-Vranceanu 3-space.

We solve the equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds via Cartan's method of equivalence. The problem separates into two branches on total space, one of which ends up with the intransitive involutive structure equations. For the transitive case, we obtain an {e}-structure on both total and base spaces.

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Elkholy and Areefi showed that in a space time, the intersection of a plane, passing through the origin, with the ligt cone, given by the equation 3 L = O, is two 2-planes 2 perpendicular to each other. In this study, instead of Eikholy-Areefi's ligt cone in a space time by dealing with the cone given by the eguation, a'j + a'^ x. . 2 '3b = O and showing that also it's intersection with 3-plane, passing through the origin, is two 2-planes perpendicular to one enother, the generalization of the aıticle of Eikholy-Areefi has heen ob tained. Furthermore, validity is proved for the sphere given by the equation 3 X, . 2 ■4 • . 2 2 '4 5/^2 +