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

Authors: Marius Munteanu and Kristopher Tapp
Journal: Proc. Amer. Math. Soc. 139 (2011), 4121-4135
MSC (2010): Primary 53C12
Published electronically: March 21, 2011
MathSciNet review: 2823057
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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.

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Additional Information

Marius Munteanu
Affiliation: Department of Mathematics, Computer Science, and Statistcs, SUNY Oneonta, Oneonta, New York 13820

Kristopher Tapp
Affiliation: Department of Mathematics, Saint Joseph’s University, 5600 City Avenue, Philadelphia, Pennsylvannia 19131

Keywords: Riemannian submersion, Lie group, good triple, doubly ruled surface
Received by editor(s): June 12, 2010
Received by editor(s) in revised form: September 28, 2010
Published electronically: March 21, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society