Ok, yeah. We obviously can't get rid of a curvature. I think SE answer [1] explains what I assumed incorrectly to mean 'locally flat'. I meant it so that you ca

Ok, yeah. We obviously can't get rid of a curvature. I think SE answer [1] explains what I assumed incorrectly to mean 'locally flat'. I meant it so that you can make a metric look like Minkowski metric at a point p, basically existence of normal coordinates. Of course second derivatives (and thus curvature) won't vanish in the neighborhood. Case closed, thank you. [1] https://physics.stackexchange.com/a/157019 https://physics.stackexchange.com/a/157019