Contributions to the Differential Geometry of Curves and Surfaces in 3-dimensional Manifolds
Bijdragen tot de differentiaalmeetkunde van krommen en oppervlakken in 3-dimensionale variëteiten
Nistor, Ana-Irina; S0208693
The present work brings new contributions to the geometry of curves and surfaces in 3-manifolds based on the recent results obtained developing two main subjects, namely the constant angle property for curves and surfaces and magnetic curves. A first topic consists in the study of constant angle surfaces and their generalization to surfaces endowed with a canonical principal direction in different 3-dimensional ambient spaces. In a few words, a surface for which its unit normal makes constant angle with a "fixed" direction is called a constant angle surface. In particular, if the projection of this direction onto the tangent plane to the surface is a principal direction, then the projection is called a canonical principal direction. We characterize and classify the surfaces endowed with a canonical principal direction in Euclidean 3-space, and in the product space H2xR, where H2 denotes the hyperbolic plane given by the Minkowski model and R is the real line, when the fixed direction is chosen to be the R-line. Another choice for the fixed direction is a Killing vector field in Euclidean 3-space and we completely classify curves and surfaces making constant angle with this direction. The second objective, treated in the last chapter, refers to the classification of normal Killing magnetic curves in the product space S2xR where S2 denotes the 2-sphere and R is the real line. We explain first how the geometric problem may be retrieved from physics. Second, in a geometric approach, we completely classify the trajectories of particles moving under the action of a Killing magnetic field in S2xR.The output of this research consists of characterization and classification theorems obtained. Moreover, illustrative examples were formulated and graphical representations were provided for a better visualization of the "geometry" of curves and surfaces.