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.

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

The main intention of this paper is to analyze integrability for the derivative formulas of the rotation minimizing frame in the Lorentz–Minkowski 3-space. As far as we know, no one has yet given a method to study their integrability in the Lorentz–Minkowski 3-space. So, we introduce the coordinate system in order to provide a tool for studying the integrability. As an application, the position vectors of some special curves having an important place in mathematical and physical research are obtained in the natural representation form. Finally, we support our work with examples.

We introduce the complete pseudo-Riemannian manifold [Formula: see text] with a conformally flat pseudo-metric satisfying Einstein’s equation. First, we get a theorem that associates the curvatures of a non-degenerate surface belonging to conformally equivalent spaces, say [Formula: see text] and [Formula: see text]. Next, we evaluate the existence of non-degenerate helicoidal surfaces in some conformally flat pseudo-spaces with pseudo-metrics corresponding to particular conformal factors (for example, a certain rotational symmetry with translational symmetry). We determine these conformal factors according to the causal characters of the axis of rotation. Right after, we get a two-parameter family of non-degenerate helicoidal surfaces with prescribed extrinsic curvature or mean curvature given by smooth functions, corresponding to both spacelike and timelike axes of rotation. As for the lightlike axis, we discuss some particular cases.

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