The Meyers-Serrin theorem on Riemannian manifolds: a survey

TL;DR

This survey explores the density of smooth functions and forms in Sobolev spaces on Riemannian manifolds, clarifying the relationship between weak covariant and partial derivatives.

Contribution

It provides a comprehensive review of the Meyers-Serrin theorem's extension to Riemannian manifolds, emphasizing the equivalence of different notions of derivatives.

Findings

Established the density of smooth functions in Sobolev spaces on Riemannian manifolds.

Clarified the equivalence between weak covariant derivatives and weak partial derivatives.

Reviewed key techniques and results related to the Meyers-Serrin theorem in geometric contexts.

Abstract

We revisit the questions of density of smooth functions, and differential forms, in Sobolev spaces on Riemannian manifolds. We carefully show equivalence of weak covariant derivatives to weak partial derivatives.