1. The theory of isoparametric submanifolds is deeply related to the theory of holonomy groups . 2. Lifts of loops about a point give rise to the holonomy group at that point. 3. If F has a holonomy group then every leaf of F is compact with finite holonomy group ." 4. If F has a holonomy group then every leaf of F is compact with finite holonomy group ." 5. See below . ) It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. 6. It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting. 7. The paper " Submanifolds with constant principal curvatures and normal holonomy groups " is a very good introduction to such theory. 8. As with the holonomy groups , the holonomy bundle also transforms equivariantly within the ambient principal bundle " P ". 9. Theorem : " Let F be a holonomy group , then all the leaves of F are compact with finite holonomy group ." 10. Theorem : " Let F be a holonomy group, then all the leaves of F are compact with finite holonomy group ."
