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Pseudo-Riemannian metrics with prescribed scalar curvature

Marc Nardmann
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Abstract

We consider the following generalisation of a well-known problem in Riemannian geometry: When is a smooth real-valued function s on a given compact n-dimensional manifold M (with or without boundary) the scalar curvature of some smooth pseudo-Riemannian metric of index q on M? We prove that this is the case for every s if 2<q<n-2, provided M admits a metric of index q at all. In fact, if 2<q<n-2, then each connected component of the space of pseudo-Riemannian metrics of index q on M contains a metric with scalar curvature s. We prove several theorems for pseudo-Riemannian metrics of index 1 or 2 as well.

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Marc Nardmann. 2004-09-22. Pseudo-Riemannian metrics with prescribed scalar curvature. https://arxiv.org/abs/math/0409435

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