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(Semi-)Riemannian geometry of (para-)octonionic projective planes. (English) Zbl 1175.53064
The main theorems of the paper under review hold for four manifolds: the octonionic projective plane \({\mathbb O}P^2\), the octonionic projective plane of indefinite signature \({\mathbb O}P^{(1,1)}\), the para-octonionic projective plane \({\mathbb O}'P^2\), and the hyperbolic dual \({\mathbb O}H^2\) of the octonionic projective plane. These manifolds are studied by means of the reduced homogeneous coordinates, which in particular are used to construct (semi-)Riemannian metrics. One proves that the aforementioned projective planes are simply connected, homogeneous, and symmetric. The Riemannian curvature tensors are computed as well. A theorem of the paper establishes the fact that the four projective planes are special Osserman manifolds with a non-trivial eigenvalue of multiplicity 7, in the sense of E. García-Río, D. N. Kupeli, and R. Vázquez-Lorenzo [Osserman manifolds in semi-Riemannian geometry. Lecture Notes in Mathematics. 1777. Berlin: Springer (2002; Zbl 1005.53040)].

53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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