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 S22 with constant angular velocity in semi-Euclidean 3-space with index two E32, 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 on S22. Furthermore, we consider some relations between semi-Riemann spaces SO(2; 1) and T1S22 such as diffeomorphism and isometry. We have obtained the system of differential equations giving geodesics of Sasaki semi-Riemann manifold (T1S22; gS). Moreover, we consider the stationary motion of a particle on S22 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 giving geodesics 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 (T1S22; gS). © TüBITAK.