An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.

In this paper, by used Frenet trihedron for a lightlike curve, the motion of Darboux rotation axis is seperated to two simultaneous rotation motions. These rotation motions are that tangent and normal vectors of lightlike curve rotate around each other. But, the angular speeds of them are different. Then, by doing the similar operations, we obtain that Darboux axis rotates around spacelike vector of Frenet trihedron of the lightlike curve and this spacelike vector rotates around Darboux axis. Consequently, we obtain the series of Darboux vectors by this way. So, simple mechanisms can be formed.

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We know that Bonnet surfaces are the surfaces which can admit at least one non-trivial isometry that preserves the principal curvatures in the Euclidean three-dimensional space. In this study, firstly, we have examined the required conditions for the canal surfaces, which are called the special swept surfaces, to be Bonnet surfaces. After that, we have defined the Bonnet canal surfaces in the Euclidean three-dimensional space and have obtained some special results for the Bonnet canal surfaces. We have studied the Bonnet canal surfaces, which will be formed, when the curve generating the canal surface is a special curve. In addition, we have given an example of the Bonnet canal surface by considering the conditions.

It is a known fact that a translation hypersurface is obtained by combination of any three curves in the 4-dimensional Euclidean space. We examine a special situation where the Gauss map of a translation hypersurface satisfies the condition G D AG where represents the Laplace operator and A is a 4 4-real matrix. Our result is that such a translation hypersurface is one of the following three hypersurfaces: the hypersurface of translation surface and a constant vector along this surface, the hyperplane, the hypersurface˙ R where˙is a translation surface.