The decomposition of tensor spaces with almost complex structure. (English) Zbl 1064.53015

Lakomá, Lenka; Jukl, Marek

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].

Reviewer: Maria Falcitelli (Bari)

MSC:

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

Keywords:

curvature tensors in almost complex manifolds