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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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##### References:
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