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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)

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