Suppose I have a (smooth) manifold with two different metrics $g,h$, where their respective scalar curvatures $R_g, R_h$ are the same. What, if anything, can I say about the relationship between the two. In particular, are they isometric? I care about complete (asymptotically flat) manifolds in particular, but would be interested in any related results.
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Of course not. Scalar curvature is a function only and carries much less information than the metric. For example, take a metric on the sphere such that its scalar curvature has two critical points, minimum and maximum, at the morth and south poles correspondently. Now smoothly slightly perturb the metric far from the poles. The scalar curvature viewed as a function remains the same, up to a diffeomorphism, because two functions on the sphere with two critical points are the same up to the diffeomorphism, iff their values at the critival points are the same, but metrics are different
answered Oct 1, 2013 at 19:50