Fixing K = −1, 0, or 1, a complete Riemannian manifold is said to have higher hyperbolic, Euclidean, or spherical rank if every geodesic admits a normal parallel field making curvature K with the geodesic. In this thesis, we establish rigidity results for three-manifolds of higher rank without a priori sectional curvature bounds. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are finite volume hyperbolic space forms. Complete three-manifolds have higher spherical rank if and only if they are spherical space forms.In addition to the rigidity results, we also provide constructions of non-homogeneous manifolds of higher hyperbolic rank of infinite volume. These examples show the necessity of the finite volume assumption in hyperbolic rank rigidity results.
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