Geodesic motions in SO 2,1

In this study,we have considered the rotational motions of a particle around the origin of the unit 2-sphere $S_2^2$ with constant angular velocity in semi-Euclidean 3-spacewithindextwo $E^3_2$, 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 on $S_2^2$. Furthermore, we consider some relations between semi-Riemann spaces $SO 2,1 $ and $T_1S_2^2$ such as diffeomorphism and isometry. We have obtained the system of differential equations giving geodesics of Sasaki semi-Riemann manifold $ T_1S_2^2,g^S $ . Moreover, we consider the stationary motion of a particle on $S_2^2$ 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 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 $ T_1S_2 2,g^S $.

Keywords:

Tangent sphere bundle, rotational group in semi-Euclidean 3-space geodesics,

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