Convergence of Vector Bundles with Metrics of Sasaki-Type

If a sequence of Riemannian manifolds, X _i , converges in the pointed Gromov–Hausdorff sense to a limit space, X _∞, and if E _i are vector bundles over X _i endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the E _i converges in the pointed Gromov–Hausdorff sense to a metric space, E _∞. The projection maps π _i converge to a limit submetry π _∞ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to (mathbb {R}^{k}/G) , where G, henceforth called the wane group, is a closed subgroup of O(k) that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms μ _i =∥?∥ _i converge to a map μ _∞ compatible with the rescaling in (mathbb {R}^{k}/G) and the (mathbb {R}) -action on E _i converges to an (mathbb {R}) -action on E _∞ compatible with the limiting norm. A natural notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the (mathbb {R}) -action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Also, necessary conditions for uniqueness of parallel translates are given in terms of the wane groups. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g., when a single compact n-dimensional manifold is rescaled to a point, the limit fiber is (mathbb {R}^{n}/H) where H is the closure of the holonomy group of the compact manifold considered. Both these examples have uniqueness of parallel translates. However, examples for non-uniqueness are also produced by looking at isolated conical singularities.