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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 .

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# Local geometry of theory of general relativity

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