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I am delving into the intricate relationship between metrics and scalar curvature in Riemannian geometry. My objective is to understand the feasibility and methods for solving the inverse problem of determining a metric $g$ given a scalar curvature $R$. I am aware that scalar curvature does not uniquely determine a metric due to the existence of distinct metrics yielding the same scalar curvature. However, I wonder if, under certain constraints or additional conditions, one might find special or particular solutions to this problem?

I found a related discussion that might illuminate aspects of this topic: Does the curvature determine the metric?

lming2
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    The existence part of the question is known as the Kazdan-Warner problem (where one actually asks for metrics with prescribed scalar curvature in a given conformal class of metrics), treated e.g. in the [new book by Andrea Malchiodi ][1].

    I guess the main reference (for existence) is the original paper [Scalar curvature and conformal deformation of Riemannian structure][2] by Kazdan and Warner.

     – ThiKu Dec 29 '23 at 09:24
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    – ThiKu Dec 29 '23 at 09:24
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    From a dimension counting argument: the scalar curvature $R$ is a single scalar function while the metric $g$ on an $n$-manifold is given by $\frac12n(n+1)$ functions, so the problem of prescribing the scalar curvature is vastly underdetermined. On the one hand, this enables the gluing arguments of Corvino-Schoen (see also extensions by Delay). On the other hand, this suggests that if you constrain your geometry so that the degree of freedom you have in the metric is only one dimensional, you may have a chance to get unique solutions. – Willie Wong Dec 29 '23 at 09:46
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    The approach taken by Kazdan and Warner is exactly in this flavour: they start with a fixed background metric $g_0$ and look for conformal metrics of the form $e^{2f} g_0$; the degree of freedom is now only that of one single scalar function $f$ and there is hope. // You can of course also consider other situations where other sorts of dimensional reduction happens, but you will have to develop the theory yourself. – Willie Wong Dec 29 '23 at 09:49
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    In addition to the two references that ThiKu provided, another older survey of these type of questions is given in Kazdan's 1985 CBMS lectures Prescribing the Curvature of a Riemannian Manifold. – Willie Wong Dec 29 '23 at 09:52

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