Based on this question from Peterson's Riemannian Geometry:
Let $(M,g)$ be an n-dimensional connected Riemannian manifold that is isometric to Euclidean space outside some compact subset $K \subset M$, i.e., $M − K$ is isometric to $\mathbb{R}^n − C$ for some compact set $C \subset \mathbb{R}^n$. If $\text{Ricg} \geq 0$, show that $M = \mathbb{R}^n$.
There are then two hints, but they basically run into the same problem in opposite directions: the first hint leads you to conclude that there are parallel 1 forms defined on $K$ (after showing $K$ is flat, and thus the whole space is flat), but I cannot go from here to conclude that there are globally defined parallel 1 forms.
The other hint tells you to extend the $n$ parallel 1 forms on $M-K$ to harmonic forms on $M$, and use Bochner's technique to conclude that these extensions are parallel. Again, I see the logic, but I am having trouble seeing how to extend the 1-forms on $M-K$ over $K$.
Any ideas?
We can assume that $[0,R]^n$ containing $K$ properly That is we have $n$-torus $T$. Since torus haz ${\bf Z}^n$ as first cohomology so there exist $n$ harmonic 1-forms If $X$ is one of them, recall $$ \Delta X_i - R_{ij}X_i =\Delta_d X_i $$
Hence $$\int_T -|\nabla X|^2 =\int_T Rc(X,X)\geq 0 $$
Hence $X$ is parallel. Bundary of $[0,R]^n$ is flat so that we have global parallel $1$-forms on ${\bf R}^n$.