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

Email:
munteam@oneonta.edu

**Kristopher Tapp**

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

Email:
ktapp@sju.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-10821-9

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