Geometric Integration by Parts and Sobolev Spaces on Vector Bundles: A Unified Global Approach
This article develops a unified and intrinsic framework for the theory of Sobolev spaces on vector bundles over Riemannian manifolds. The analytical core of our approach is an explicit higher-order geometric integration by parts formula, which characterizes the formal adjoint of the covariant derivative as a global differential operator. This identity is established on arbitrary Riemannian manifolds with boundary, without assuming completeness or compactness. While first-order integration by parts identities are classical, explicit higher-order formulas with precise boundary terms are rarely stated in the literature. As applications of this framework, we recover the classical Meyers--Serrin theorem on arbitrary manifolds and, in the compact case, the Sobolev embedding and Rellich--Kondrachov compactness theorems, providing direct and self-contained proofs. At the end of this work we also stablish a Green Formula and we use it to stablish norm equivalence in Sobolev Spaces on vector bundles with closed manifold as base space and the Bochner laplacian operator. As a corollary we show that, in case of trivial vector bundles this equivalence reduces to a well known (but non proved rigorously in literature) result for closed manifolds and the Laplace-Beltrami operator. By emphasizing intrinsic global arguments and sharp local-to-global norm equivalence estimates, rather than ad hoc coordinate patching, this work offers a transparent and accessible foundation for the study of Sobolev spaces on vector bundles, suitable for researchers in global analysis, differential geometry, and partial differential equations.