Curvas en los espacios 3-dimensionales pseudo-riemannianos de curvatura constante

Laburpena

ABSTRACT OF THE THESIS 1. Objectives The main aim of the thesis is to obtain new results in the study of curves of Bertrand, rectifying curves and slant helices in the three-dimensional spaces of constant curvature. We present the most important properties for each family of curves, bearing in mind the type of the curve (Frenet, pseudo-null or null curve). We obtain its natural equations (in terms of its curvatures and arc parameter), and we obtain many properties and characterization theorems. For each family we introduce some surfaces (conic surface, helix surface) which allow us its geometric integration. 2. Research methodology The methodology followed in preparing this thesis is the standard one in any research work in mathematics. A bibliographic review of the field of the thesis. Once we have known the literature, study and comprehension of the different published papers which determine the base of such a topic. The study, at the same time, of the useful tools and techniques in order to understand different works. Study of the problem by using the studied material, reflecting progresses in research papers to publish in international journals. Active participation in the geometry seminar which carries out in the research group in which I am included. Discussions of the problems and progresses with the advisor of the thesis. 3. Results or conclusions The first contribution in the Bertrand's curve topic is the extension of its definition to 3-dimensional model spaces, and this allows us to obtain its natural equation. We characterize plane curves and ordinary hélices as the only Bertrand curves with infinite Bertrand conjugates. We present several characterization theorems for pseudo-null and null Bertrand curves. We caracterize the functionals, depending only on the curvature, whose phase space consists only of Bertrand curves. We also present new results in the study and integration of Bertrand curves in pseudo-Euclidean spaces. We perform the extension of the rectifying curve definition and its connection with conic surfaces. We show that the rectifying curves are the geodesics of conic surfaces. We find the parameterizations of these geodesics and the relationship of their curvatures with their arc parameter (natural equation). We characterize the developable rectifying surfaces. Specifically, these surfaces are conical surfaces if, and only if, the curve generating it is rectifying. The main theorem offers a geometric integration for the rectifying curves from spherical curves. Finally, we have a theorem of characterization of the rectifying curves as minimums of a given function and a result that relates the rectifying curves that are curves of Bertrand with the generalized conical helices. Finally, we extend the definition of the slant helices. We obtain that there are no slant helices in the hyperbolic and de-Sitter spaces. We determine the natural equations of slant helices. We introduce the concept of helix surface and prove that they are flat surfaces. Then we characterize the flat surfaces that are helix surfaces. Finally, we achieve the geometric integration of the slant helices and characterize them as the geodesics of the helix surfaces. We conclude with the geometric integration of the pseudo-null and null slant helices. The thesis contributes to the study of the curves in the 3-dimensional pseudo-Riemannian spaces of constant curvature, being this a subject that is between the differential geometry and the Riemannian geometry. Our contribution has been published in 8 research papers in international journals of high impact, as can be seen in references [LO12], [LO13a], [LO13b], [LO14], [LO15], [LO16a], [LO16b] and [LO17] of the memory being presented.