Abstract
In [16], Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for Riemannian submersions from nonnegatively curved spaces even without the strict positive curvature assumption and irrespective of the particular metric.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berestovskii V. N., Guijarro L.: A metric characterization of Riemannian submersions, Ann. Global Anal. Geom. 18, 577–588 (2000)
R. Bott and L. Tu Differential forms in algebraic topology, Graduate Texts in Mathemtics, 82 (1982).
D. Gromoll and G. Walschap Metric Foliations and Curvature, Progr. in Math., 268, Birkhauser Verlag (2009).
L. Guijarro, Petersen P.: Rigidity in non-negative curvature. Ann. Sci. École Norm. Sup. 30, 595–603 (1997)
L. Guijarro, T. Schick and G. Walschap Bundles with spherical Euler class, Pacific Journal of Mathematics 207 (2002), 377–391.
L. Guijarro and G. Walschap Twisting and nonnegative curvature metrics on vector bundles over the round sphere, Journal of Differential Geometry 52 (2000), 189–202
L. Guijarro and G. Walschap Transitive holonomy groups and rigidity in nonnegative curvature, Mathematische Zeitschrift 237(2001), 251–257.
L. Guijarro and G. Walschap When is a Riemannian submersion homogeneous?, Geom. Dedicata 125 (2007), 47–52.
R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fiber bundle, Proc. of the AMS, 11 (1960), 236–242.
M. Kerr and K. Tapp A note on quasi-positive curvature conditions, preprint, http://arxiv.org/abs/1211.3897
J. McCleary, A User’s Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics (2000).
Y. Shi The dual foliation of some singular Riemannian foliations, preprint.
N. Steenrod, The topology of fiber bundles, Princeton University Press, (1951).
G. Walschap, Nonnegatively curved manifolds with souls of codimension 2, J. Diff. Geometry 27 (1998), 525–537.
G. Walschap, Metric foliations and curvature, J. Geom. Anal. 2 (1992), 373–381.
B. Wilking, A duality theorem for Riemannian foliations in nonnegative sectional curvature, Geom. Funct. Anal. 17 (2007), 1297–1320.
Author information
Authors and Affiliations
Corresponding author
Additional information
P. Angulo-Ardoy and L. Guijarro were supported by research grants MTM2011-22612 from the Ministerio de Ciencia e Innovación (MCINN) and MINECO: ICMAT Severo Ochoa project SEV-2011-0087.
Rights and permissions
About this article
Cite this article
Angulo-Ardoy, P., Guijarro, L. & Walschap, G. Twisted submersions in nonnegative sectional curvature. Arch. Math. 101, 171–180 (2013). https://doi.org/10.1007/s00013-013-0550-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-013-0550-z