Almost Hermitian 6-manifolds revisited. (English) Zbl 1075.53036

Let \((M,g)\) be a Riemannian manifold equipped with a \(G\)-structure. A connection is called a characteristic connection \(\nabla^c\) of the \(G\)-structure if it is a \(G\)-connection with totally skew-symmetric torsion \(T^c\), i.e., \(T^c\) corresponds to a 3-form. For a Riemannian naturally reductive space \(M = G_1/G\) the characteristic connection of the corresponding \(G\)-reduction coincides with the canonical connection of the reductive space and moreover \(T^c\) and the curvature tensor \(R^c\) are parallel. This parallelism does not hold in general but \(\nabla^cR^c = 0\) is an important condition.
In this paper the authors study the case where the \(G\)-structure is given by an almost Hermitian structure \(J\) and they focus on the 4- and 6-dimensional case. For these cases they derive conditions for the existence of characteristic connections and then study these cases by using a decomposition of the space of 3-forms. For \(\dim M = 4\) they obtain that \((M^4,g,J)\) admits a characteristic connection if and only if it is a Hermitian manifold and moreover, \(T^c\) is parallel if and only if \((M^4,g,J)\) is a generalized Hopf manifold (i.e., the Lee form \(\delta \Omega \circ J\) is parallel with respect to the Levi Civita connection, \(\Omega\) being the Kähler form).
Next, they consider the 6-dimensional spaces where they discuss their decompositions and combine them with the known one leading to the well-known classification of almost Hermitian structures into 16 types. Then they specialize to the special classes of nearly Kähler spaces and of \(G_1\)-spaces, respectively. For the first class they indicate a proof of a result of Kirichenko stating that the characteristic connection exists and has parallel torsion \(T^c\). For the second case they prove that if \((M,g,J)\) is of type \(G_1\) (a class containing the naturally reductive spaces) then it admits a unique characteristic connection. Furthermore, they study the case where its torsion \(T^c\) is parallel and prove that there are only two types of semi-Kähler spaces: the twistor space of a 4-dimensional self-dual Einstein space and the invariant Hermitian structure on SL\((2,\mathbb C)\), respectively. Finally, all naturally reductive semi-Kähler spaces with small isotropy group of \(\nabla^c\) are determined.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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