×

zbMATH — the first resource for mathematics

(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)].

MSC:
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Besse, A.L., Manifolds all of whose geodesics are closed, (1978), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0387.53010
[2] Brown, R.; Gray, A., Riemannian manifolds with holonomy group spin(9), (), 41-59
[3] Freudenthal, H., Oktaven, ausnahmegruppen und oktavengeometrie, (1951), Mathematisch Intituut der Rijksuniversiteit te Utrecht · Zbl 0056.25905
[4] Freudenthal, H., Oktaven, ausnahmegruppen und oktavengeometrie, Geom. dedicata, 19, 1, 7-63, (1985) · Zbl 0573.51004
[5] Garcia-Rio, E.; Kupeli, D.; Vazquez-Lorenzo, R., Osserman manifolds in semi-Riemannian geometry, Lecture notes in mathematics, vol. 1777, (2002), Springer-Verlag Berlin · Zbl 1005.53040
[6] Gilkey, P.B., Geometric properties of natural operators defined by the Riemann curvature tensor, (2001), World Scientific Press · Zbl 1007.53001
[7] Jordan, P., Über eine nicht-desarguessche ebene projektive geometrie, Abh. math. sem. univ. Hamburg, 16, 74-76, (1949) · Zbl 0034.38103
[8] Mostow, G.D., Strong rigidity of locally symmetric spaces, Annals of mathematics studies, vol. 78, (1973), Princeton University Press/University of Tokyo Press Princeton, NJ/Tokyo · Zbl 0265.53039
[9] Moufang, R., Alternativkörper und der satz vom vollständigen vierseit, Abh. math. sem. univ. Hamburg, 9, 207-222, (1933) · JFM 59.0551.03
[10] Nikolayevsky, Y., Osserman conjecture in dimension \(n \ne 8, 16\), Math. ann., 331, 3, 505-522, (2005) · Zbl 1075.53016
[11] Osserman, R., Curvature in the eighties, Amer. math. monthly, 97, 8, 731-756, (1990) · Zbl 0722.53001
[12] Rosenfeld, B., Geometry of Lie groups, (1997), Kluwer Academic Publishers Group Dordrecht · Zbl 0867.53002
[13] Wolf, J., Spaces of constant curvature, (1974), Publish or Perish, Inc. Boston, MA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.