We know that when we derive the Geodesic equation, we want to actually understand the geometrical meaning of the Riemann tensor. We see from the geodesic equation that the second derivative of the deviation vector is proportional to the Riemann tensor. This tells that whether our spacetime is curved or not.

Now let us assume that the covariant derivative along a geodesic curve of any arbitrary vector (except tangent vector) $A^i$ vanishes. Now we assume that the vector field along $A^i$ is the deviation vector field, and after evaluating the geodesic deviation of $A^i$ along this geodesic curve, we get 0 because this deviation vector is assumed to be parallel transported. So, this implies that the Riemann tensor vanishes, hence the spacetime is flat. So from this, we can also conclude that if covariant derivative of any arbitrary vector vanishes along any geodesic curve then my spacetime is flat. I know my conclusion is wrong, but I am not getting where I am wrong in my argument.

Is there something wrong where I am assuming that vector $A^i$ whose covariant derivative vanishes is the deviation vector? Can I assume this?


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