Abstract
We find constraints on the extent to which O’Neill’s horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp’s theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.
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Aloff S., Wallach N.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)
Bazaikin Y.V.: On one family of 13-dimensional closed Riemannian positively curved manifolds. Sib. Math. J. 37, 1219–1237 (1996)
Berger M.: Les variétés riemanniennes à courbure positive. Bull. Soc. Math. Belg. 10, 88–104 (1958)
Cheeger J.: Some examples of manifolds of nonnegative curvature. J. Differ. Geom. 8, 623–628 (1973)
Dearricott, O.: A 7–manifold with positive curvature. Duke Math. J. (to appear)
Eschenburg, J.-H.: Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen. Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], 32, pp. vii+177. Universität Münster, Mathematisches Institut, Münster (1984)
Eschenburg J.-H.: Inhomogeneous spaces of positive curvature. Differ. Geom. Appl. 2(2), 123–132 (1992)
Eschenburg J.-H.: New examples of manifolds with strictly positive curvature. Invent. Math. 66, 469–480 (1982)
Gray A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967)
Gromoll D., Meyer W.: An exotic sphere with nonnegative sectional curvature. Ann. Math. 100(2), 401–406 (1974)
Gromoll D., Walschap G.: Metric Foliations and Curvature. Birkhäuser, Basel (2009)
Guijarro L., Walschap G.: The metric projection onto the soul. Trans. Am. Math. Soc. 352(1), 55–69 (2000)
Guijarro L., Walschap G.: When is a Riemannian submersion homogeneous?. Geom. Dedic. 125, 47–52 (2007)
Grove, K., Verdiani, L., Ziller, W.: A positively curved manifold homeomorphic to T 1 S 4. Geom. Funct. Anal. http://arxiv.org/abs/0809.2304. (to appear)
Grove, K., Ziller, W.: Lifting group actions and nonnegative curvature. Trans. Am. Math. Soc. http://arxiv.org/abs/0801.0767. (to appear)
Munteanu M.: One-dimensional metric foliations on compact Lie Groups. Mich. Math. J. 54, 23–25 (2006)
O’Neill B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)
Petersen, P., Wilhelm, F.: An exotic sphere with positive sectional curvature. preprint. http://arxiv.org/abs/0805.0812
Petersen, P., Wilhelm, F.: Some principals for deforming nonnegative curvature preprint. http://arxiv.org/abs/0908.3026
Strake M., Walschap G.: Σ-flat manifolds and Riemannian submersions. Manuscr. Math. 64, 213–226 (1989)
Tapp K.: Flats in Riemannian submersions from Lie groups. Asian J. Math. 13(4), 459–464 (2009)
Wallach N.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96, 277–295 (1972)
Wilking B.: A duality theorem for Riemannian foliations in nonnegative sectional curvature. Geom. Funct. Anal. 17, 1297–1320 (2007)
Wilking B.: Manifolds with positive sectional curvature almost everywhere. Invent. Math. 148(1), 117–141 (2002)
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Pro, C., Wilhelm, F. Flats and submersions in non-negative curvature. Geom Dedicata 161, 109–118 (2012). https://doi.org/10.1007/s10711-012-9696-2
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DOI: https://doi.org/10.1007/s10711-012-9696-2