Suppose I have a R^3 manifold that goes into R^3 charts, if that is possible. The manifold has curvature and is Riemannian and has a metric. I want to eliminate all curvature in R^3 charts, so I want to add another dimension to the manifold, I would extract all the curvature information from the manifold and deposit it into this new 4th dimension. I would probably also have to add some kind of topology or map to the new dimension describing the curvature. But I would have flat R^3 chart. Is this possible at all?