Hello guys . Through all the analysis of theory of general relativity we used what so called Manifolds Manifolds as we know are topological spaces that resemble ( look like) euclidean space locally at tiny portion of space And an euclidean space is the pair ( real coordinate space R^n , dot product ), And any euclidean space is flat space, So manifolds locally are flat , do not have curvature locally But solutions of GR's equations show the manifolds are not flat locally even they are locally look like euclidean space . My question is that if the space is curved , Then the Curvature tensor does not depend on the chosen local real coordinate space ( system ) , is it ?! Thanks .