Abstract
We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution establishing a general Myers theorem for quaternionic contact manifolds of positive horizontal Ricci curvature. We introduce the category of almost Einstein quaternionic manifolds, characterized by the vanishing of one component of the torsion for the Biquard connection. Under the assumption of non-positive horizontal sectional curvatures, we show that the universal cover of any complete almost Einstein quaternionic contact manifold is either \(\mathbb {R}^{h+3}\) or \(\mathbb {R}^h \times \mathbb {S}^3\).
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References
Aubin, T.: Métriques riemanniennes et courbure. J. Differ. Geom. 4, 383–424 (1970)
Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque 265 (2000)
Davidov, J., Ivanov, S., Minchev, I.: The twistor space of a quaternionic contact manifold. Q. J. Math 63(4), 873–890 (2012)
de Andres, L., Fernandez, M., Ivanov, S., Ugarte, L., Vassilev, D.: Quaternionic Kaehler and Spin(7) metrics arising from quaternionic contact Einstein structures. Annali di matematica Pura ed Applicata (2012). doi:10.1007/s10231-012-0276-8
Duchemin, D.: Quaternionic contact structures in dimension 7. Ann. Inst. Fourier 56(3), 851–885 (2006)
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom 17(2), 255–306 (1982)
Hebda, J.J.: Curvature and focal points in Riemannian foliations. Indiana Univ. Math. J. 35(2), 321–331 (1986)
Hladky, R.K.: Connections and curvature in sub-Riemannian geometry. Houston J. Math. 38(4), 1107–1134 (2012)
Hladky, R.K.: Bounds for the first eigenvalue of the horizontal Laplacian on postively curved sub-Riemannian manifolds. Geom. Dedicata 164, 155–177 (2013)
Ivanov, S., Minchev, I, Vassilev, D.: Quaternionic contact Einstein manifolds. arXiv:1306.0474v1
Ivanov, S., Minchev, I., Vassilev, D.: Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem. J. Eur. Math. Soc. 12, 1041–1067 (2010)
Ivanov, S., Minchev, I., Vassilev, D.: The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group. Ann. Sc. Norm. Super Pisa Cl. Sci X I(5), 635–652 (2012)
Ivanov, S., Minchev, I., Vassilev. D.: Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem. Mem. Am. Math. Soc. 231(1086) (2014, to appear)
Ivanov, S., Petkov, A., Vassilev, D.: The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold. J. Geom. Anal. 24(2), 756–778 (2014)
Ivanov, S., Vassilev, D.: Conformal quaternionic contact curvature and the local sphere theorem. J. Math. Pure Appl. 93, 277–307 (2010)
Johnson, D.L., Whitt, L.B.: Totally geodesic foliations. J. Differ. Geom. 15(2), 225–235 (1980)
O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–460 (1966)
Reinhart, B.L.: Foliated manifolds with bundle-like metrics. Ann. Math. (2) 69(1), 119–132 (1959)
Zhu, S.-H.: On open three-manifolds of quasi-positive Ricci curvature. Proc. Am. Math. Soc. 120(2), 569–572 (1994)
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Hladky, R.K. The topology of quaternionic contact manifolds. Ann Glob Anal Geom 47, 99–115 (2015). https://doi.org/10.1007/s10455-014-9437-x
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DOI: https://doi.org/10.1007/s10455-014-9437-x