Gravitational Curvature and Its Asymptote

As we know the whole universe is covered by curved space, so everything works by disturbing curve in space, even a smallest part of the universe cannot perform its work without creating curvature in space. And so gravitational pull also construct wide curvature in space. If gravity can be defined as the curvature of space rather than a force of attraction, why does not a bullet shot out of a gun, say perpendicular to the Earth's crust, and a ball thrown on the same trajectory (but obviously at a much slower speed) follow the same curve? If gravity is actually curved space and if falling objects are following simply the natural curves of space why does each object have its own curve?