This post is related to this and this post.
It is known that on a complete Riemannian manifold, the space $C^\infty_c(M)$ is generally not dense in the Sobolev spaces $W^{k, p}(M)$ ($1 \leq p < \infty$). However, a sufficient criterion for this to hold is what one calls bounded geometry of order $k-2$, meaning that the first covariant derivatives of the curvature tensor are uniformly bounded and that the injectivity radius is positive. A standard reference seems to be Eichhorn's book "Global Analysis on Open Manifolds".
This means that in general, the orthogonal complement of the space $W_0^{k,2}(M)$, the closure of $C^\infty_c(M)$ inside $W^{k,2}(M)$ is non-empty. However, I find it very hard to get an intuition for that. Therefore my first question.
Q1: What is an explicit example of a non-trivial function $f \in W_0^{2,2}(M)^\perp \subset W^{2,2}(M)$ on some explicit Riemannian manifold $M$?
Here I have in mind some nice $2$-dimensional manifold $M$ embedded in $\mathbb{R}^3$ or so. In particular, I would be interested in an example where $M$ has bounded curvature but the injectivity radius goes to zero as I have no intuition for why the latter would be important at all (I tried different surfaces of revolution but couldn't find an answer).
My second question is regarding the source I already mentioned: In Eichhorn's book, the result (i.e. if $M$ has bounded curvature of order $k-2$, then $C^\infty_c(M)$ is dense in $W^{k,p}(M)$) is not proved, but rather referred to some older paper of his, which is not available online. So far I couldn't get my hands on a copy. Therefore:
Q2: Is a proof of this result somewhere available online, or written down in some newer book?
