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Geometric realizations of Hermitian curvature models. (English) Zbl 1205.53016
A complex curvature model is a quadruplet \({\mathcal C}= (V,\langle,\rangle,J,A)\), where \((V,\langle,\rangle)\) is a Euclidean vector space, \(J\) a Hermitian complex structure on \(V\) and \(A\) an algebraic curvature tensor on \(V\).
In [Differ. Geom. Appl. 27, No. 6, 696–701 (2009; Zbl 1191.53017)], the authors and G. Weingart proved that, given a complex curvature model \({\mathcal C}\), there exist an almost Hermitian manifold \(M\) and a point \(p\in M\) such that \({\mathcal C}\) is geometrically realized by \(M\) at \(p\). Furthermore, the manifold \(M\) can be chosen to have constant scalar curvature and constant \(*\)-scalar curvature. This result gives a complete answer to the geometric realization problem for complex curvature models.
The paper under review deals with the analogous problem concerning the so-called Hermitian curvature models, namely, the complex curvature models \({\mathcal C}= (V,\langle,\rangle,J,A)\), where \(A\) is an algebraic curvature tensor which satisfies the Gray identity:
\[ \begin{aligned} A(x,y,z,w) &+ A(Jx,Jy,Jz, Jw)=A(Jx,Jy,z,w) + A(x,y,Jz,Jw)+ A(Jx,y,Jz,w)\\ & + A(x,Jy,z,Jw)+A(Jx,y,z,Jw) + A(x,Jy,Jz,w),\end{aligned}\tag{1} \] for any \(x, y,z,w\in V\).
Note that the Riemannian curvature \(R_p\) at any point \(p\) of a Hermitian manifold satisfies (1).
The main result of this paper is the following.
Theorem 1. Let \({\mathcal C}= (V,\langle,\rangle, J,A)\) be a complex curvature model. Then \(A\) satisfies (1) if and only if there exist a Hermitian manifold \(M\) and a point \(p\in M\) such that \({\mathcal C}\) is geometrically realized by \(M\) at \(p\).
Furthermore, the manifold \(M\) in Theorem 1 can be chosen to have constant scalar and \(*\)-scalar curvatures, and the point \(p\) can be chosen in such way that the fundamental form is closed at \(p\). In particular, it follows that the Kähler condition at a single point does not imply additional curvature restrictions.

MSC:
53B20 Local Riemannian geometry
53B35 Local differential geometry of Hermitian and Kählerian structures
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