Kowalski, Oldřich; Vanhecke, Lieven Classification of five-dimensional naturally reductive spaces. (English) Zbl 0555.53024 Math. Proc. Camb. Philos. Soc. 97, 445-463 (1985). A general conjecture suggested by the authors says that, on a naturally reductive homogeneous Riemannian space, (M,g), every two I(M)-invariant differential operators commute. This conjecture was proved earlier for dimensions 3 and 4 by the same authors. The aim of the present paper is to prove the conjecture for dimension five via the explicit classification. (An interesting feature of the classification itself is that the most general classes depend on several real parameters and one rational parameter.) Cited in 1 ReviewCited in 22 Documents MSC: 53C30 Differential geometry of homogeneous manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) Keywords:commutative spaces; reductive homogeneous Riemannian space; invariant differential operators; classification PDFBibTeX XMLCite \textit{O. Kowalski} and \textit{L. Vanhecke}, Math. Proc. Camb. Philos. Soc. 97, 445--463 (1985; Zbl 0555.53024) Full Text: DOI References: [1] Helgason, Differential Geometry and Symmetric Spaces (1962) [2] D’Atri, Naturally Reductive Metrics and Einstein Metrics on Compact Lie groups 215 (1979) [3] DOI: 10.1307/mmj/1029001423 · Zbl 0317.53045 · doi:10.1307/mmj/1029001423 [4] D’Atri, J. Differential Geometry 9 pp 251– (1974) [5] Tricerri, Homogeneous Structures on Riemannian Manifolds 83 (1983) · doi:10.1017/CBO9781107325531 [6] DOI: 10.1112/blms/15.1.35 · Zbl 0521.53048 · doi:10.1112/blms/15.1.35 [7] Lichnerowicz, Ann. Sc. Ecole Norm. Sup. 81 pp 341– (1964) [8] Kowalski, Generalized Symmetric Spaces. 805 (1980) · Zbl 0431.53042 · doi:10.1007/BFb0103324 [9] Kowalski, Classification of Generalized Symmetric Riemannian Spaces of Dimension n 5 (1975) · Zbl 0343.53033 [10] Kobayashi, Foundations of Differential Geometry II (1969) [11] DOI: 10.2307/2372398 · Zbl 0059.15805 · doi:10.2307/2372398 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.