On 2d gravity and parallel transport field equation on quantum sphere

In this work we have obtained the exact quantum expressions for the compenents of the Levi Cevita connection, the Ricci tensor and the scalar curvature, generalizing those of \cite{CTZX08} for a spherical surface via the noncommutative Moyal star product, and we have established equations describing quantum effect on the geodesic flow equation or auto parallel fields equations. These later are solved for the zero and the first orders of the quantum parameter $\alpha$ when $y$-symmetry is assumed. We expressed the general system of equations in terms of Fourier modes indexed by the integer $p=1,2,...,\infty$ to understand the interdependence of $y$ modes oscillations.