Some New Properties of Surfaces Generated by Null Cartan Curves

In this paper, some special types of surfaces with null Cartan base curve are investigated. The generating lines of the surfaces are chosen as a linear combination of Cartan frame fields with non-constant differentiable functions. Firstly, the surfaces whose generating lines have the same direction of Cartan frame fields B, N and T are examined respectively. As a special case, Gaussian and mean curvatures of one parameter family of Bertrand curves of a given null Cartan curve and the singular points of this type of surface are stated. Furthermore, an example is also stated to explain the obtained results. Then, the surfaces with null Cartan base curve are investigated where generating lines lie on the planes spanned by {N, B}, {T, B} and {T, N}, respectively. Finally, some differential geometric properties of these surface are given mainly in three different cases.