Are Dark Matter and space curvature how scientists escape from geocentrism?

No. The fact that the Earth moves is evident from the existence of parallax. That is to say, we can actually use instruments to measure the extent to which stars in the sky shift back and forth over the course of each year as the Earth travels around the Sun, thereby viewing those stars from slightly different positions and in slightly different directions. This was how the distances between stars were first measured. Moreover, with more advanced instruments, the change in Doppler shift from light from other stars can also be measured as the Earth's direction of travel changes over the course of each year. Geocentrism just doesn't hold up against these observations, which are not dependent on either dark matter or general relativity but purely on high school geometry and wave physics.