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

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.

💡 Research Summary

The paper presents a comprehensive and intrinsically geometric framework for Sobolev spaces on vector bundles over arbitrary Riemannian manifolds, without imposing completeness or compactness assumptions. The authors begin by fixing a smooth finite‑rank vector bundle (E\to M) equipped with a fiber metric (h_E) and a compatible connection (\nabla_E). Using the induced connection on the tensor bundles (T^{0,s}(TM)\otimes E), they define the strong Sobolev space (H^{m,p}(E)) as the completion of smooth sections with respect to the norm (|u|{m,p} = (\sum{s=0}^m\int_M |\nabla^s u|^p,d\lambda_g)^{1/p}), and the weak Sobolev space (W^{m,p}(E)) as the set of (L^p) sections whose weak covariant derivatives up to order (m) exist in the distributional sense.

The analytical heart of the work is Theorem 1, a higher‑order integration‑by‑parts formula for covariant derivatives. For compactly supported sections (F) and (G) of appropriate tensor type, the identity \