Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Velázquez-Mendoza Carlos Daniel, Sandoval-Romero María de los Ángeles

TL;DR

This work addresses the global theory of Sobolev spaces on vector bundles over Riemannian manifolds by introducing a higher-order geometric integration by parts formula that yields an explicit, intrinsic characterization of the formal adjoint of the covariant derivative. It provides self-contained, global proofs of the Meyers-Serrin density theorem on general manifolds and, for compact manifolds, streamlined Sobolev embedding and Rellich-Kondrashov results, all built from sharp local-to-global norm equivalences. The approach reframes Sobolev spaces in intrinsic geometric terms, avoiding reliance on elliptic regularity or torus reductions, and yields a robust foundation for global analysis on bundles with potential PDE applications. Overall, the paper unifies and extends key Sobolev-space results to vector bundles via a purely geometric, globally coherent framework.

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.

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

TL;DR

This work addresses the global theory of Sobolev spaces on vector bundles over Riemannian manifolds by introducing a higher-order geometric integration by parts formula that yields an explicit, intrinsic characterization of the formal adjoint of the covariant derivative. It provides self-contained, global proofs of the Meyers-Serrin density theorem on general manifolds and, for compact manifolds, streamlined Sobolev embedding and Rellich-Kondrashov results, all built from sharp local-to-global norm equivalences. The approach reframes Sobolev spaces in intrinsic geometric terms, avoiding reliance on elliptic regularity or torus reductions, and yields a robust foundation for global analysis on bundles with potential PDE applications. Overall, the paper unifies and extends key Sobolev-space results to vector bundles via a purely geometric, globally coherent framework.

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.
Paper Structure (8 sections, 11 theorems, 152 equations)

This paper contains 8 sections, 11 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation and Conventions
  4. Sobolev Spaces
  5. The Geometric Integration by Parts Formula
  6. Local-to-Global Estimates and Structural Theorems
  7. Global Approximation Results
  8. Sobolev Embeddings and the Rellich--Kondrashov Theorem

Key Result

Lemma 1

Let $(M,g)$ be a Riemannian manifold and let $\nabla$ denote a connection on $TM$. Let $E\longrightarrow M$ be a smooth vector bundle with a connection $\nabla^{E}$. We denote by $\nabla$ the induced connection on the tensor product bundle $T^{(k,l)}(TM)\otimes E$, defined uniquely by the Leibniz ru for all $X \in \Gamma(TM)$, $F\in \Gamma(T^{(k,l)}(TM))$, and $\sigma \in \Gamma(E)$. Let $(E_{1},\

Theorems & Definitions (25)

  • Definition 1: The space $H^{m,p}$
  • Definition 2: Weak Covariant Derivative and the space $W^{m,p}$
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1: Integration by Parts for Higher-Order Derivatives
  • proof
  • Corollary 1: Local Expression of the Formal Adjoint
  • proof
  • Lemma 2
  • ...and 15 more