## 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.

### MSC:

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

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