Sobolev Spaces – Density of Smooth Functions on Manifolds

Question

Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have been defined as the collection of locally integrable functions whose weak derivatives (w.r.t to the Levi-Civita connection) are in L^p. For compact manifolds, or for manifolds with bounded geometry and positive injectivity radius (like $\mathbb{R}^n$), I can see why these two definitions are equivalent. Are they equivalent for general noncompact manifolds?

Answer 1

According to pages 14 and 15 of:

• MR2343536 Reviewed Eichhorn, Jürgen Global analysis on open manifolds. Nova Science Publishers, Inc., New York, 2007. x+644 pp. ISBN: 978-1-60021-563-6; 1-60021-563-7 (Reviewer: Yuri A. Kordyukov)

The poof is given in:

• MR1066741 Reviewed Eichhorn, Jürgen Elliptic differential operators on noncompact manifolds. Seminar Analysis of the Karl-Weierstrass-Institute of Mathematics, 1986/87 (Berlin, 1986/87), 4–169, Teubner-Texte Math., 106, Teubner, Leipzig, 1988. (Reviewer: Steven Rosenberg)

The results are:

The closure of smooth functions with compact support, the closure of smooth function in the Sobolev space, and the Sobolev space are all different in general on open Riemannian manifolds. If the manifold is of bounded geometry (of order $k$) then all these spaces coincide up to Sobolev order $k+2$. This holds even for sections of vector bundles. Thus on compact manifolds all these spaces coincide also.