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The decomposition of tensor spaces with almost complex structure. (English) Zbl 1064.53015
Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 145-150 (2004).
Given an almost complex manifold \((M^n,J)\), the authors compute a \(J-d\) decomposition of each tensor field \(A\) of type \((1,3)\) on \(M^n\) which satisfies for any vector fields the following conditions: \(X\), \(Y\), \(Z\).
\(A(X,Y,Z)= -A(Y,X,Z)\), \(\underset{X,Y,Z}{}\sigma\, A(X,Y,Z)= 0\), \(A(X,Y,Z)= A(JX,JY, Z)\),
\(\text{tr\,}{\mathcal A}(X, Y)= 0\),
where \({\mathcal A}(X,Y,)(Z)= A(X,Y,Z)\).
If \(n\geq 4\), each tensor field \(A\) can be uniquely expressed as a sum of a tensor field \(B\), which is \(J\)-traceless, and of five tensor fields depending on the Ricci tensor associated with \(A\).
In particular, considering \(A\) as the Riemannian curvature of a Kähler manifold, the corresponding \(J\)-traceless tensor \(B\) is the holomorphically-projective curvature.
For the entire collection see [Zbl 1034.53002].

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D15 Almost contact and almost symplectic manifolds