\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {chapter}{\numberline {2}Background}{3}
\contentsline {section}{\numberline {2.1}The metric projection onto a soul}{3}
\contentsline {section}{\numberline {2.2}The normal holonomy group of a soul}{5}
\contentsline {section}{\numberline {2.3}The ideal boundary}{6}
\contentsline {section}{\numberline {2.4}Basic properties of Riemannian submersions}{7}
\contentsline {chapter}{\numberline {3}Conditions for nonnegative curvature on vector bundles}{8}
\contentsline {section}{\numberline {3.1}Curvature at the soul}{8}
\contentsline {section}{\numberline {3.2}Curvature near the soul}{12}
\contentsline {section}{\numberline {3.3}Connection metrics on vector bundles}{14}
\contentsline {section}{\numberline {3.4}The warping tensor}{18}
\contentsline {section}{\numberline {3.5}Warped connection metrics on vector bundles}{21}
\contentsline {section}{\numberline {3.6}Summary of conditions for nonnegative curvature}{26}
\contentsline {chapter}{\numberline {4}Riemannian submersions with compact holonomy}{28}
\contentsline {section}{\numberline {4.1}Consequences of compact holonomy}{28}
\contentsline {section}{\numberline {4.2}Vertizontal curvature decay}{30}
\contentsline {section}{\numberline {4.3}A soul with noncompact normal holonomy}{31}
\contentsline {section}{\numberline {4.4}Measuring size in the holonomy of a vector bundle}{32}
\contentsline {chapter}{\numberline {5}Volume growth bounds}{35}
\contentsline {section}{\numberline {5.1}Previous results}{35}
\contentsline {section}{\numberline {5.2}An upper bound on volume growth}{36}
\contentsline {section}{\numberline {5.3}Bounded vertical Jacobi fields}{38}
\contentsline {section}{\numberline {5.4}A lower bound on volume growth}{39}
\contentsline {chapter}{\numberline {6}Bounded Riemannian submersions}{41}
\contentsline {section}{\numberline {6.1}A bound on the folding of the fibers}{41}
\contentsline {section}{\numberline {6.2}Measuring size in the holonomy of a submersions}{44}
\contentsline {section}{\numberline {6.3}The ideal boundary is determined by one fiber}{45}
\contentsline {section}{\numberline {6.4}Two finiteness theorems for Riemannian submersions}{47}
\contentsline {chapter}{\numberline {7}Finiteness theorems for fiber bundles}{50}
\contentsline {section}{\numberline {7.1}Introduction}{50}
\contentsline {section}{\numberline {7.2}A finiteness theorem for vector bundles}{52}
\contentsline {section}{\numberline {7.3}A finiteness theorem for principal bundles}{55}