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Almost Hermitian geometry on six dimensional nilmanifolds. (English) Zbl 1072.53010

A. Gray and L. M. Hervella [Ann. Mat. Pura Appl. 123, 35-58 (1980; Zbl 0444.53032)] have classified almost Hermitian manifolds by obtaining characterizations of sixteen different classes. They considered the vector space \(W\) of tensors of type \((0,3)\) of a \(2n\)-dimensional vector space \(V\) with the same symmetry properties as the covariant derivative of \(F\) with respect to the Levi-Civita connection \(\nabla\) associated to \(g\) and studied the decomposition of this space as a direct sum of subspaces invariant and irreducible under the natural action of the unitary group \({\text U}(n,\mathbb R)\) on \(W\). They found four terms \(W_ i\) in this decomposition. In this paper, the authors describe the space \(\mathcal Z\) of all left-invariant almost complex structures \(J\) on a real 6-dimensional Lie group \(G\) which are compatible with a prescribed metric \(g\) and a fixed orientation. When \(G\) is nilpotent and \(\Gamma\) is a discrete subgroup the authors give a general method for classifying the structures \(J\) of \(\mathcal Z\) in such a way that the compact manifold \(M=\Gamma\backslash G\) belongs to one of the 16 Gray-Hervella classes. Since \(\mathcal Z\) and \(\mathbb C{\text P}^3\) are isomorphic, \(\mathcal Z\) is visualized as a tetrahedron, in which the edges and faces represent projective subspaces \(\mathbb {\text P}^1\) and \(\mathbb C{\text P}^2\). Moreover, they describe the fundamental 2-form of an invariant almost Hermitian structure on a 6-dimensional Lie group in terms of the action of \(\text{SO}(4)\times \text{U}(1)\) on a complex projective 3-space which leads to combinatorial description of the classes of almost Hermitian structures on the Iwasawa and other nilmanifolds. A complete description of the 16 Gray-Hervella classes in terms of faces, edges and vertices of the tetrahedron is obtained.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
51A05 General theory of linear incidence geometry and projective geometries
17B30 Solvable, nilpotent (super)algebras

Citations:

Zbl 0444.53032
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References:

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