Lotta, A. On the fundamental group of compact homogeneous manifolds carrying an invariant fat distribution. (English) Zbl 1367.53046 Arch. Math. 108, No. 6, 625-628 (2017). The main result of the paper is the following: Let \(M=G/H\) be a homogeneous space of a compact Lie group \(G\). If \(M\) admits a \(G\)-invariant fat distribution, then \(\pi_1(M)\) is finite. The meaning of fatness is explained in the paper. Roughly, it is related to a property of the Levi-Tanaka form of the distribution. The key point in the proof is to show that the \(G\)-invariance of the fat distribution forces the Ricci tensor to be positive, so that (by homogeneity) Myer’s theorem implies that \(\pi_1(M)\) is finite. Reviewer: Andreas Arvanitoyeorgos (Patras) MSC: 53C30 Differential geometry of homogeneous manifolds 58A30 Vector distributions (subbundles of the tangent bundles) 32V05 CR structures, CR operators, and generalizations Keywords:fat distribution; strongly bracket generating distribution; Levi-Tanaka form; homogeneous CR manifold PDFBibTeX XMLCite \textit{A. Lotta}, Arch. Math. 108, No. 6, 625--628 (2017; Zbl 1367.53046) Full Text: DOI Link References: [1] I. Agricola, The Srní lectures on non-integrable geometries with torsion, Arch. Math. (Brno) 42 (2006), suppl., 5-84. · Zbl 1164.53300 [2] D. M. Almeida, On fat sub-Riemannian symmetric spaces in codimension three, Differential Geom. Appl. 24 (2006), 178-190. · Zbl 1093.53033 · doi:10.1016/j.difgeo.2005.09.010 [3] H. Azad, A. Huckleberry, and W. Richthofer, Homogeneous CR-manifolds, J. Reine Angew. Math. 358 (1985), 125-154. · Zbl 0553.32016 [4] D. Burns and S. Shnider, Spherical hypersurfaces in complex manifolds, Invent. Math. 33 (1976), 223-246. · Zbl 0357.32012 · doi:10.1007/BF01404204 [5] A. Kaplan and M. Subils, On the equivalence problems for bracket-generating distributions. Hodge theory, complex geometry, and representation theory, Contemp. Math. 608, Am. Math. Soc., Providence, RI, 2014, 157-171. · Zbl 1303.53036 [6] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0175.48504 [7] A. Spiro, Groups acting transitively on compact CR manifolds of hypersurface type, Proc. Amer. Math. Soc. 128 (2000), 1141-1145. · Zbl 0961.32031 · doi:10.1090/S0002-9939-99-05113-8 [8] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221-263. · Zbl 0609.53021 · doi:10.4310/jdg/1214440436 [9] N. Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23-84. · Zbl 0409.17013 · doi:10.14492/hokmj/1381758416 [10] F. Tricerri and L. Vanhecke, Naturally reductive homogeneous spaces and generalized Heisenberg groups, Compositio Math. 52 (1984), 389-408. · Zbl 0551.53028 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.