Geodesic motions in SO(2,1)

Abstract

In this study, we have considered the rotational motions of a particle around the origin of the unit 2-sphere S2 2 with constant angular velocity in semi-Euclidean 3-space with index two E3 2 , namely geodesic motions of SO(2, 1). Then we have obtained the vector and the matrix representations of the spherical rotations around the origin of a particle onS 2 2 . Furthermore, we consider some relations between semi-Riemann spaces SO(2, 1) and T1S2 2 such as diffeomorphismand isometry. We have obtained the system of differential equations giving geodesics of Sasaki semi-Riemann manifold(T1S 2 2 , gS ) . Moreover, we consider the stationary motion of a particle on S2 2 corresponding to one parameter curve of SO(2, 1), which determines a geodesic of SO(2, 1). Finally, we obtain the system of differential equations givinggeodesics of the semi-Riemann space (SO(2, 1), h) and we show that the system of differential equations giving geodesics of Riemann space (SO(2, 1), h) is equal to that of (T1S2 2 , gS ).