Integration by parts formula and Sobolev embeddings on vector bundles: an intrinsic geometrical 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–Kondrashov compactness theorems, providing direct and self-contained proofs. At the end of this work, we also establish Green’s formula and use it to establish norm equivalence in Sobolev spaces on vector bundles over closed manifolds for the Bochner Laplacian. As a corollary, we recover the classical norm equivalence for the Laplace–Beltrami operator on closed manifolds and provide a complete proof which, to the best of our knowledge, does not appear explicitly in the standard literature. 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.