Some specijal types of curves, frames and surfaces in Minkowski spaces

Teorija Rimanovih i semi-Rimanovih podmnogostrukosti je jedna od najinteresantnijih oblasti u klasičnoj i savremenoj diferencijalnoj geometriji. Pored toga, diferencijalna geometrija podmnogostrukosti u prostorima Minkovskog je oblast istraživanja koja je poslednjem periodu dala mnoge nove rezultate, naročito u teoriji svetlosnih podmnogostrukosti. U ovoj doktorskoj disertaciji predstavljene su neke specijalne vrste krivih, repera i površi u prostorima Minkovskog. Dobijene su eksplicitne parametarske jednačine prostornih rektifikacionih krivih u prostoru Minkovskog E31 čija je projekcija na prostornu, vremensku ili svetlosnu ravan normalna kriva. Takođe su date eksplicitne parametarske jednačine prostornih normalnih krivih u istom prostoru čija je projekcija na svetlosnu ravan u odnosu na izabranu skrin distribuciju rektifikaciona W-kriva.The theory of Riemannian and semi-Riemannian submanifolds is one of the most interesting areas in classical and modern differential geometry. Besides, differential geometry of submanifolds in Minkowski spaces is the reasearch area that recently has given many new results in investigations, in particular in the theory of lightlike submanifolds. In this thesis, we present some special types of curves, frames and surfaces in Minkowski spaces. We obtain explicit parameter equations of the spacelike rectifying curves in Minkowski space R 3 1 whose projection onto spacelike, timelike and lightlike plane is a normal curve. We also obtain explicit parameter equations of the spacelike normal curves in the same space whose projection onto lightlike plane with respect to a chosen screen distribution, is a rectifying W-curve. In this thesis it is proved that there are no null Mannheim curves in Minkowski space. It is also proved that the only pseudo null Mannheim curves in Minkowski space are pseudo null straight lines and pseudo null circles. The notion of Mannheim curves is further generalized by introducing the generalized null Mannheim curves in Minkowski space-time. Such curves and their generalized Mannheim mate curves are characterized in terms of their curvature functions. In particular, the relations between their frames are obtained. In this thesis we also define the generalized partially null Mannheim curves and the generalized pseudo null Mannheim curves in Minkowski space-time. We prove that there are no non-geodesic generalized partially null Mannheim curves, by considering the cases when the corresponding mate curve is spacelike, timelike, null Cartan, partially null, or pseudo null Frenet curve