Although "Spacetime Curvature is Gravity" is the first statement that is told to an infant by the "popular science", I believe that really really misrepresents what GR has to say. I am posting this question to confirm whether my understanding is appropriate or not.
I have the following arguments against "Spacetime Curvature is Gravity":
Technically speaking, we mean gravitational field by gravity in the classical physics, and thus, according to the correspondence principle, the quantity used to describe gravity in a modern theory should, in the classical limit, become the good old (Newtonian) gravitational field. Christoffel symbols are the quantities that do so and not the Reimann curvature tensor. So, how come the curvature of spacetime can be called gravity?
More importantly, the curvature of spacetime can be zero and we can still have non-zero Christoffel symbols. Since the vanishing or non-vanishing nature of Christoffel symbols is what determines whether the laws of Physics take the special relativistic form or not in a given frame, we can most certainly have gravity in the absence of the curvature of spacetime.
It is argued at some places that it is the curvature tensor that determines whether there is any "true gravity" or not. Specifically, if the curvature is zero everywhere in the spacetime, one can always have a transformation that takes her from a generic coordinate system to a global Minkowskian coordinate system - in other words, one can globally gauge-away the gravity. Okay, great - but just the fact that you can gauge away the gravity doesn't mean that it doesn't really exist in those frames as well in which the laws of physics don't have a special relativistic form. One has got to accept in GR that the existence or otherwise of gravity is purely a frame-dependent fact. I think this is the most important insight of general relativity that the existence of gravity itself is a relativistic fact - it is perfectly fine that one frame sees gravity and the other doesn't. The famous "elevator experiment" - where an elevator, in the deep empty space (say, with zero curvature), is getting pulled by some rope - illustrates the very fact that wholeheartedly accepting that gravity is truly existent in the frame of the accelerated elevator is the way to do Physics in the accelerated frames. I agree that one can, in the cases where there is no curvature at all, do away with special relativity by cleverly chosing the inertial frame or by keep on transforming his equations referring to the inertial frame. But there is no other way to do physics in a non-inertial frame without keep referring to an other frame than accepting that gravity exists in non-inertial frames even if the spacetime is Minkowskian. And honestly, if one is willing to assert that just because gravity can be gauged away, gravity doesn't really exist then one should actually say that gravity doesn't exist locally even if the curvature is non-zero. But certainly, such an assertion is absurd.
So, shouldn't we say "Christoffel Symbols are Gravity" rather than "Spacetime Curvature is Gravity"?