On types of non-integrable geometries. (English) Zbl 1079.53041

Bureš, Jarolím (ed.), The proceedings of the 22nd winter school “Geometry and physics”, Srní, Czech Republic, January 12–19, 2002. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 71, 99-113 (2003).
A G-structure on a Riemannian manifold is said to be integrable if it is preserved by the Levi-Civita connection. In the presented paper, the following non-integrable G-structures are studied: SO(3)-structures in dimension 5; almost complex structures in dimension 6; G\(_2\)-structures in dimension 7; Spin(7)-structures in dimension 8; Spin(9)-structures in dimension 16 and F\(_4\)-structures in dimension 26. G-structures admitting an affine connection with totally skew-symmetric torsion are characterized. It is known [S. Ivanov, {Connections with torsion, parallel spinors and geometry of {Spin(7)}-manifolds}, math.dg/0111216] that any Spin(7)-structure admits a unique connection with totally skew-symmetric torsion. In this paper, it is proved that under weak conditions on the structure group this is the only geometric structure with that property. Moreover, the automorphisms group of non-integrable geometric structures are studied.
For the entire collection see [Zbl 1014.00011].


53C10 \(G\)-structures
53C05 Connections (general theory)
53C27 Spin and Spin\({}^c\) geometry
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