Group norms and their degeneration in the study of Parallelism
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Endowing total spaces of vector bundles over Riemannian manifolds with a Riemannian structure sets them within the realm of Gromov's Theory of Convergence. The particular choice of Riemannian metric is a generalization of the one studied by Sasaki on tangent bundles. In this work, the static and the dynamic properties of said bundles are studied. Here static means the metric and differential geometric properties of the interplay between the Riemannian metrics of the base and the total space. Differential geometrically, the fibers are known to be flat and totally geodesic. Metrically, it is shown that their departure from convexity is controlled quite explicitly by the concept of holonomic spaces. A holonomic space is a triple $(V,H,L)$, where $V$ is a normed vector space, $H$ is a group of norm preserving linear maps, and $L$ is a group norm, together with a convexity assumption. In the particular geometric setting, $V$ is a fixed fiber of a vector bundle, $H$ is the holonomy group at that fiber, and $L$ is a geometric group norm, the length-norm, obtained by looking at the smallest loop that generates a given holonomy element. The degenerations of these group-norms are fundamental to determining the dynamic properties. It is also seen that by restricting the class of maps to geodesic Riemannian maps, the Sasaki metric construction renders the tangent bundle a metric functor. The dynamic perspective is to analyze the convergence of these metrics of Sasaki type under the Gromov-Hausdorff topology. A pre-compactness result is obtained under the assumption of a uniform upper bound on rank. Furthermore, the limiting spaces possess a surprisingly rich structure. Limits of Sasaki-type metrics are submetries over the limit of their bases and retain a notion of re-scaling and a compatible norm (understood here as a distance to zero of sorts). The fibers of which are conical are worst: in fact, their topology is that of a quotient of Euclidean space by a compact group of orthogonal transformations. This group, called the wane group, is essentially obtained by looking at limits of holonomy elements with waning length-norm; it depends on the base point, and thus the limits in general fail to be locally trivial. These groups will further play a r\^ole for the uniqueness problem of a limiting notion of parallelism, also introduced here. The length-norm studied here had been overlooked before perhaps due to its lack of continuity with respect to the standard Lie group topology on the holonomy groups. However, tautologically, a group norm is continuous with respect to the metric topology induced by itself. This topology, seemingly artificial, also has some of the nice properties one should require a topological transformation group to have; even certain wrong way inheritance is exhibited. Overall, this work dwells upon the interactions between the metric properties and the algebraic nature of vector bundles, as well as their possible degeneration in a limiting process.