# Riemannian Submersions And Lie Groups

 dc.contributor.advisor Grove, Karsten en_US dc.contributor.author Jimenez, William en_US dc.date.accessioned 2005-08-03T15:23:17Z dc.date.available 2005-08-03T15:23:17Z dc.date.issued 2005-05-27 en_US dc.identifier.uri http://hdl.handle.net/1903/2648 dc.description.abstract The principal topic of this thesis is the study of Riemannian submersions and Riemannian foliations of Lie groups. We classify Riemannian submersions of tori $T$ with left invariant (hence bi-invariant) metric. In this case, the fiber through the identity $F_e$ of the torus is a subgroup of $T$, and the submersion itself must be the coset projection $T \rightarrow T/F_e$. Then we characterize Riemannian submersions $f: G \rightarrow H$ where $G$ and $H$ are Lie groups with left invariant metric. These are homomorphisms if and only if the basic lift of a left invariant vector field in $H$ is a left invariant vector field in $G$. A similar theorem is proven to characterize antihomomorphisms. We then study a Ricatti equation associated with a Riemannian foliation, giving new proofs of several results of Walschap. We also give a relationship that must hold between the fiber dimension and base dimension of a Riemannian submersion, thus partially answering a question of Wilhelm. We then turn to the study of homogeneous Riemannian foliations. First we give a sufficient condition for a Riemannian foliation to be homogeneous, and use it to show that any Riemannian submersion of a compact Lie group with totally geodesic connected fibers is homogeneous if the fiber through the identity is a Lie subgroup. This is a result related to work of Ranjan \cite{Ranjan2}. Finally, we discuss the homogeneity of flat foliations of symmetric spaces of compact type, culminating in a proof that compact simple Lie groups with bi-invariant metric do not admit Riemannian foliations of codimension 1 with closed leaves. en_US dc.format.extent 281146 bytes dc.format.mimetype application/pdf dc.language.iso en_US dc.title Riemannian Submersions And Lie Groups en_US dc.type Dissertation en_US dc.contributor.publisher Digital Repository at the University of Maryland en_US dc.contributor.publisher University of Maryland (College Park, Md.) en_US dc.contributor.department Mathematics en_US dc.subject.pqcontrolled Mathematics en_US
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