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Zbl 1072.53010
Garbiero, Sergio; Abbena, Elsa; Salamon, Simon
Almost Hermitian geometry on six dimensional nilmanifolds. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No. 1, 147-170 (2001). ISSN 0391-173X

{\it A.~Gray} and {\it 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 ${\text U}(n,\Bbb R)$ on $W$. They found four terms $W\sb i$ in this decomposition. In this paper, the authors describe the space $\Cal Z$ 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 $\Cal Z$ in such a way that the compact manifold $M=\Gamma\backslash G$ belongs to one of the 16 Gray-Hervella classes. Since $\Cal Z$ and $\Bbb C{\text P}^3$ are isomorphic, $\Cal Z$ is visualized as a tetrahedron, in which the edges and faces represent projective subspaces $\Bbb {\text P}^1$ and $\Bbb C{\text P}^2$. 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.
[Andrew Bucki (Edmond)]

MSC 2000:: *53C15 Geometric structures on manifolds
53C55 Complex differential geometry (global)
51A05 General theory of linear incidence geometry
17B30 Solvable, nilpotent Lie algebras

Citations: Zbl 0444.53032

Cited in: Zbl 1239.53071

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