Totally Geodesic Foliations and Doubly Ruled Surfaces in a Compact Lie Group - Mathematics > Differential GeometryReport as inadecuate




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Abstract: For a Riemannian submersion from a simple compact Lie group with abi-invariant metric, we prove the action of its holonomy group on the fibers istransitive. As a step towards classifying Riemannian submersions with totallygeodesic fibers, we consider the parameterized surface induced by lifting abase geodesic to points along a geodesic in a fiber. Such a surface is -doublyruled- it is ruled by horizontal geodesics and also by vertical geodesics.Its characterizing properties allow us to define -doubly ruled parameterizedsurfaces- in any Riemannian manifold, independent of Riemannian submersions. Weinitiate a study of the doubly ruled parameterized surfaces in compact Liegroups and in other symmetric spaces by establishing several rigidity theoremsand by providing several examples with unexpected properties.



Author: Marius Munteanu, Kristopher Tapp

Source: https://arxiv.org/



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