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Geometric realizations of curvature models by manifolds with constant scalar curvature. (English) Zbl 1191.53017
Let \(V\) be a real vector space of dimension \(n\) and let \(A\) be an algebraic curvature tensor on \(V\). We recall that \({\mathcal M} = (V,\langle\;,\;\rangle, A)\) is a curvature model if \(A\) is an algebraic curvature tensor on \(V\) and if \(\langle\;,\;\rangle\) is a non-degenerate bilinear form of signature \((p,q)\) on \(V\).
It is well known that given a curvature model \(\mathcal M\), there exist a real analytic pseudo-Riemannian manifold \(M\) and a point \(p\) of \(M\) such that \(M\) has a curvature model isomorphic to \(\mathcal M\). The authors extend this result to the class of manifolds with constant scalar curvature showing that any pseudo-Riemannian curvature model can be geometrically realized by a pseudo-Riemannian manifold with constant scalar curvature. Moreover, they prove that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and constant \(*\)-scalar curvature.
Reviewer: Anna Fino (Torino)

53B20 Local Riemannian geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text: DOI
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