Geometry of some curves in the generalized Cartan-Vranceanu manifolds

The aim of this thesis is to characterize the curve couples of the biharmonic curves in the Cartan-Vranceanu manifold. In the first chapter, some preliminary information concerning Riemann manifold and biharmonic curves which will be used in the subsequent chapters is given. In the second chapter, some results concerning Cartan-Vranceanu manifolds, biharmonic curves on these manifolds in the literature are given. Two chapters that come after the second chapter, are completely original. In the third chapter, characterizations of curve couples of the biharmonic curves in the 3−dimensional Cartan-Vranceanu manifolds are given. These couples are evolute, involute, Bertrand and Mannheim couples respectively. In the fourth chapter, the results that handled in previous chapter are generalized to curve couples of the biharmonic curves in the (2n+1)−dimensional Cartan-Vranceanu manifolds. In addition to this, the order of evolute and involute curve couple of a biharmonic curve in the (2n + 1)−dimensional Cartan-Vranceanu manifolds are studied.