The Geometric Properties Of The Conformal Curves

In this thesis, we study the associated conformal curves in 5-dimensional Lorentz space with the curves in 3-dimensional space. The n-dimensional Mobius group is isomorphic to the (n+2)-dimensional Lorentz group which preserves the sign of the first coordinate. Spheres in 3-dimensional space can be associated with the points in 5-dimensional Lorentz space. Using the cluster of osculationg spheres, curves in 3-dimensional space can be mapped to the curves in 5-dimensional Lorentz space. And a 5-dimensional frame{T1,T2,T3,T4,T5} can be established.In this paper, we discuss some geometric properties of conformal curves with this frame.1. Using this conformal frame, following the ideas of the general helix in classical differential geometry, we discuss the conformal geometric properties of curve when it satisfies T2(t)·u=c, where u is a unit constant vector, and c is a constant.2. Using this conformal frame, following the ideas of the general helix in classical differential geometry, we discuss the conformal geometric properties of curve when it satisfies T3(t)·u=c, where u is a unit constant vector, and c is a constant. We obtain a first order ordinary differential equation of the ratio between conformal curvature and conformal torsion.3. Using this confomal frame, we obtain the Cesaro identical condition and the condition of fixed point of vector in 5-dimensional Lorentz space.