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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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
  • 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
  • © Copyright 2011 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 4121-4135
  • MSC (2010): Primary 53C12
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10821-9
  • MathSciNet review: 2823057