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Curvature of spacetime (note the singular) of stationay universe is none since spacetime has to be flat on the Noether theorem that requires flat (or Minkowski, which means the same thing) spacetime for the conservation of energy to hold.

The lack of curvature of spacetime does not reqire all of 10 curvatures of spacetime (note the plurar) to disappear since they, being non zero, may combine together in flat spacetime as it may happen in stationary universe where some curvatures of spacetime together with others may combine in a flat (Minkowski) spacetime.