I am a bit confused as to how the Equivalence Principle implies a curved spacetime. Or if it doesn’t imply a curved spacetime, then what exactly makes it necessary to have a curved space time?
I could very well have local inertial frames in flat spacetime in arbitrary coordinates. Particles on which only gravity acts could follow curved paths in flat spacetime and their trajectories would be straight locally.
So what exactly forces the manifold to be curved? How exactly is the result that our spacetime should be curved reached?
Why not just study gravity as a background force field on a flat space time?
Edit after the answer:
Why can we just suppose that the world lines of freely falling particles are curved and not straight but in a flat spacetime?
Then the divergence of the geodesics which is explained by the Riemann Tensor would not need that explanation. Two falling bodies could very well come near each other because their worldliness would be curved in the flat spacetime.
Why don’t we model gravity as an external force field on a flat spacetime just like other force fields?