On totally geodesic foliations and doubly ruled surfaces in a compact Lie group

Munteanu, Marius; Tapp, Kristopher

doi:10.1090/S0002-9939-2011-10821-9

On totally geodesic foliations and doubly ruled surfaces in a compact Lie group
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by Marius Munteanu and Kristopher Tapp PDF

Proc. Amer. Math. Soc. 139 (2011), 4121-4135 Request permission

Abstract:

For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove that the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is “doubly ruled” (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define “doubly ruled parameterized surfaces” in any Riemannian manifold, independent of Riemannian submersions. We initiate a study of the doubly ruled parameterized surfaces in compact Lie groups and in other symmetric spaces by establishing several rigidity theorems and constructing examples.

References

Richard H. Escobales Jr., Riemannian submersions with totally geodesic fibers, J. Differential Geometry 10 (1975), 253–276. MR 370423
Richard H. Escobales Jr., Riemannian submersions from complex projective space, J. Differential Geometry 13 (1978), no. 1, 93–107. MR 520604
Luis A. Florit, Doubly ruled submanifolds in space forms, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 689–701. MR 2300625
Detlef Gromoll and Karsten Grove, A generalization of Berger’s rigidity theorem for positively curved manifolds, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 2, 227–239. MR 911756, DOI 10.24033/asens.1530
Detlef Gromoll and Karsten Grove, The low-dimensional metric foliations of Euclidean spheres, J. Differential Geom. 28 (1988), no. 1, 143–156. MR 950559
Detlef Gromoll and Gerard Walschap, The metric fibrations of Euclidean space, J. Differential Geom. 57 (2001), no. 2, 233–238. MR 1879226
Karsten Grove, Geometry of, and via, symmetries, Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000) Univ. Lecture Ser., vol. 27, Amer. Math. Soc., Providence, RI, 2002, pp. 31–53. MR 1922721, DOI 10.1090/ulect/027/02
D. Hilbert and S. Cohn-Vossen, Geometry and the imagination, Chelsea Publishing Co., New York, N. Y., 1952. Translated by P. Neményi. MR 0046650
M. Kerin and K. Shankar, Riemannian submersions from simple compact Lie groups, preprint, 2009.
Marius Munteanu, One-dimensional metric foliations on compact Lie groups, Michigan Math. J. 54 (2006), no. 1, 25–32. MR 2214786, DOI 10.1307/mmj/1144437436
Akhil Ranjan, Riemannian submersions of compact simple Lie groups with connected totally geodesic fibres, Math. Z. 191 (1986), no. 2, 239–246. MR 818668, DOI 10.1007/BF01164028
Kristopher Tapp, Nonnegatively curved vector bundles with large normal holonomy groups, Proc. Amer. Math. Soc. 136 (2008), no. 1, 295–300. MR 2350416, DOI 10.1090/S0002-9939-07-08983-6
Kristopher Tapp, Flats in Riemannian submersions from Lie groups, Asian J. Math. 13 (2009), no. 4, 459–464. MR 2653712, DOI 10.4310/AJM.2009.v13.n4.a2
Burkhard Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001), no. 2, 281–295. MR 1826371, DOI 10.1007/PL00005801
Burkhard Wilking, A duality theorem for Riemannian foliations in nonnegative sectional curvature, Geom. Funct. Anal. 17 (2007), no. 4, 1297–1320. MR 2373019, DOI 10.1007/s00039-007-0620-0