Geometric Integration by Parts and Sobolev Spaces on Vector Bundles: A Unified Global Approach

Abstract: This article develops a unified framework for the theory of Sobolev spaces on vector bundles over Riemannian manifolds. The analytical core of our approach is a rigorous higher-order geometric integration by parts formula, which characterizes the formal adjoint of the covariant derivative. This identity is established for arbitrary manifolds, requiring no assumptions on completeness or compactness. While these results are fundamental to global analysis, explicit and direct proofs are often elusive in the literature or rely on overly sophisticated machinery that overshadows the underlying geometry. To bridge this gap, we establish sharp local-to-global norm equivalence estimates and provide streamlined, self-contained proofs for the Meyers-Serrin theorem on general manifolds, as well as the Sobolev embedding and Rellich-Kondrashov theorems for the compact case. By prioritizing intrinsic global arguments over ad hoc coordinate patching, this work provides a modern and accessible foundation for the study of Sobolev spaces on bundles.