Sharp Concentration Bounds for Vector Bundle-Valued Statistics on Manifolds

Many geometric-statistics and manifold-learning pipelines produce bundle-valued observations (for example, tangent vectors) that lie in different fibers. Forming empirical averages, therefore, requires transporting data to a common reference fiber, which can introduce curvature- and holonomy-driven effects. We study transported empirical means for a measurable section s: M → E of a rank-k vector bundle (E, π, M) equipped with a bundle metric and a compatible metric connection, and we derive finite-sample, dimension-free concentration bounds in the reference fiber E_{x₀}. Using sharp Hilbert-space inequalities due to Pinelis, we obtain Hoeffding- and Bernstein-type tail bounds controlled by a uniform per-sample bound B and a variance proxy σ². When minimizing geodesics are not unique, we isolate a deterministic holonomy ambiguity term Δ_{hol} and bound it in terms of bundle curvature and loop geometry; for the specific case E = T S²_{r} we give sharp area-based formulas. The resulting bias–variance decomposition provides practical uncertainty quantification for bundle-valued averaging and clarifies when curvature-driven transport ambiguity is negligible relative to sampling error.