Browsing by Subject "Walker manifold"

Now showing 1 - 3 of 3

Open Access
Bertrand and Mannheim curves in three-dimensional Walker manifolds
(2024-09) Naciri, Asmaa
We review the basic concepts of space curves, including curvature and torsion. We examine special curves such as Mannheim and Bertrand in a three-dimensional Euclidean space. We define Walker manifolds which are pseudo-Riemannian manifolds with a parallel null distribution. Then we compute Christoffel symbols and Levi-Civita connection components for an arbitrary three-dimensional Walker manifold. Finally, we derive the curvature and torsion of a regular curve on a three-dimensional Walker manifold. Then, we investigate necessary and sufficient conditions for Mannheim curves in a strict three-dimensional Walker manifold. Moreover, we also prove necessary and sufficient conditions of Bertrand curves in a three-dimensional Walker manifold.
Open Access
Geodesics of three-dimensional walker manifolds
(2016-07) Büyükbaş Çakar, Gökçen
We review some basic facts of Lorentzian geometry including causality and geodesic completeness. We depict the properties of curves and planes in threedimensional Minkowski space. We deffne the Walker manifolds, that is, a Lorentzian manifold which admits a parallel degenerate distribution. We calculate the Christoffel symbols and Levi-Civita connection components, Riemann curvature and Ricci curvature components for an arbitrary three-dimensional Walker manifold and strictly Walker manifold. Finally, we derive the geodesic equations of a three-dimensional Walker manifold and investigate the geodesic curves in it, particularly the ones with a constant component. We prove that any straight line with a constant third component is a geodesic in any Walker manifold with the causality depending on its second component. We prove that the existence of a geodesic in a Walker manifold with a linear third component implies that the manifold is strict. We also show that any three-dimensional Walker manifold is geodesically complete.
Open Access
Minimal surfaces on three-dimensional Walker manifolds
(2017-06) Berani, Erzana
Lorentzian Geometry has shown to be very useful in a wide range of studies including many diverse research elds, especially in the theory of general relativity and mathematical cosmology. A Walker manifold descends from the structure of Lorentzian manifolds which is characterized by admitting a parallel degenerate distribution. In the present thesis, we investigate and derive the equations of minimal surfaces on three-dimensional Walker manifolds, with a particular interest on those surfaces which are represented by the graph of a smooth function. Our study is closely related with (Lorentzian) isothermal coordinates which provide an easier approach for deriving such equations, and they are locally de ned for any surface on the underlying manifold. By using the well-known property of vanishing mean curvature for minimal surfaces, together with the geometric restrictions posed by the chosen coordinates, we obtain a class of graphs of functions which are minimal under certain conditions on the corresponding function.