Let's say that, from my point of view, another observer is accelerating. Now, from his point of view, he is standing still: all he feels is an overall fictitious force of gravity, which is just a consequence of the the spacetime curvature at that point (caused by stress-energy tensor at the same point).

Now, if we accept this geometric view, It seems like acceleration cannot really exist in spacetime: every object keeps following a geodesic. When some external observer thinks that an object is accelerating, he should say instead that it is following a "curved geodesic".

Is this correct?