In a recent works Liu and Wang (2008; 2007) study the Mannheim partner curves in the three dimensional space. In this paper, we extend the theory of the Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim. It is shown that, every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of it. Moreover, we obtain that the Mannheim offset of developable ruled surface is constant distance from it. Finally, examples are also given.

We dene a Bertrand-B curve in Riemannian manifold M such that thereexists an isometry \phi of M, that is, \left( \phi \circ \beta \right) (s)=X\left( s,t(s)\right) and the binormal vector of another curve \beta is the paralel vector of binormal vector of \alpha at corresponding points. We obtain the conditions of existence of a Bertrand-B curve in the event E^3, S^3 and H^3 of M. The rst of our main results is that the curve \alpha in E^3 is a Bertrand-B curve if and only if it is planar. Second one, we prove that the curve \alpha with the curvatures \epsilon _{1},\epsilon _{2} in S^3 is a Bertrand-B curve if and only if it is satises \epsilon _{1}^{2}+\epsilon _{2}^{2}=1. Finally, we state that there not exists a Bertrand-B curve in H^3.

We analyze integrability for the derivative formulas of the rotation minimizing frame in the Euclidean 3-space from a viewpoint of rotations around axes of the natural coordinate system. We give a theorem that presents only one component of the indirect solution of the rotation minimizing formulas. Using this theorem, we find a lemma which states the necessary condition for the indirect solution to be a steady solution. As an application of the lemma, the natural representation of the position vector field of a smooth curve whose the rotation minimizing vector field (or the Darboux vector field) makes a constant angle with a fixed straight line in space is obtained. Also, we realize that general helices using the position vector field consist of slant helices and Darboux helices in the sense of Bishop.

ÖZET: Bu çalışmada, 3-boyutlu Minkowski uzayında Bishop çatısına göre null olmayan regle yüzeylerin Gauss ve ortalama eğrilikleri hesaplanmış ve regle yüzeylerin sabit Gauss ve ortalama eğriliğe sahip olmaları için gerekli koşullar elde edilmiştir. Ayrıca regle yüzeylerin açılabilir ve minimal olması için denklemlerinin hangi formda olması gerektiği gösterilmiştir. Buna ilaveten teğet vektör alanı tarafından üretilen sabit ortalama eğrilikli null olmayan regle yüzeylerin dayanak eğrisinin Bishop slant helis olduğu tespit edilmiştir.

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