Surfaces in E³ invariant under a one parameter group of isometries of E³

Abstract

We develop a convenient surface theory in E 3 in order to apply it to the class of the surfaces invariant under a one-parameter group of isometries of E 3 . In this way we derive intrinsic characterizations along with several results of subclasses of this class of surfaces that satisfy certain preassigned properties. In the process all results are also effortlessly derived. Among these subclasses are those with surfaces; of constant mean curvature, of constant Gaussian curvature, isothermic, with constant difference or ratio of the principal curvatures.

SURFACES IN 𝔼3MAKING CONSTANT ANGLE WITH KILLING VECTOR FIELDS

“…We obtain the complete classification of these surfaces, which includes: half-planes having z-axis as boundary, rotational surfaces around z-axis, right cylinders over logarithmic spirals and Dini's surface. Regarding Dini's surface as a particular kind of helicoidal surfaces, on this broader class of surfaces many studies were developed, let us mention [4,15,29,30] and references therein. Classification and characterization results were obtained having as starting point the fact that helicoidal surfaces are applicable upon rotational surfaces.…”