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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
DOI: https://doi.org/10.1090/S0002-9939-2011-10821-9
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
Email: munteam@oneonta.edu

Kristopher Tapp
Affiliation: Department of Mathematics, Saint Joseph’s University, 5600 City Avenue, Philadelphia, Pennsylvannia 19131
MR Author ID: 630309
Email: ktapp@sju.edu

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