Some Characterization of Spiral Surfaces

In this paper we will study special spiral surfaces in the three dimensional Euclidean space and we give some characterization of these surfaces. More specifically, we investigate the Chang-Yau operator acting on the Gauss map of spiral surfaces. We also give some results about canonical vector field of these surfaces, i.e., we study incomperssibility of canonical vector field in two types of spiral surfaces. Moreover, we give some necessary conditions for a spiral surface to be a Weingarten surface. Existence of umbilical point is another problem that we investigate about it for a special case of spiral surfaces of the first type.

On Bäcklund and Ribaucour transformations for hyperbolic linear Weingarten surfaces

We consider Bäcklund transformations for hyperbolic linear Weingarten surfaces in Euclidean 3-space. The composition of these transformations is obtained in the Permutability Theorem that generates a 4-parameter family of surfaces of the same type. Since a Ribaucour transformation of a hyperbolic linear Weingarten surface also gives a 4-parameter family of such surfaces, one has the following natural question. Are these two methods equivalent, as it occurs with surfaces of constant positive Gaussian curvature or constant mean curvature? By obtaining necessary and sucient conditions for the surfaces given by the two procedures to be congruent.The analytic interpretation of the geometric results is given in terms of solutions of the sine-Gordon equation.

Hyperbolic Weingarten surfaces

Weingarten surfaces which can be represented locally as solutions to second order hyperbolic partial differential equations are examined in this paper. In particular, the geometry of the families of curves corresponding to characteristics on these surfaces is investigated and the relationships of these curves with other curves on the surface such as asymptotic lines and lines of curvature are explored. It is shown that singularities in the lines of curvature, i.e. umbilic points, correspond to singularities in the families of characteristics, and that lines of curvature are non-characteristic curves. If there is a linear relation between the Gaussian and mean curvatures and real characteristics exist, then the characteristics form a Tchebychef net on the corresponding Weingarten surface.