When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ structure on the long (half-)line is compatible with at least continuum-many $C^\omega$ (i.e., real-analytic) structures, something which I seemed to remember, but I wasn't sure and couldn't remember a reference. The Wikipedia article on the long line states that “any given $C^\infty$ structure can be extended in infinitely many ways to different $C^\omega$ (=analytic) structures (which are pairwise non-diffeomorphic as analytic manifolds)”, but the reference given, viꝫ. the 1960 paper by H. Kneser & M. Kneser, “Reell-analytische Strukturen der Alexandroff-Halbgeraden und der Alexandroff-Geraden” (Archiv der Mathematik 11 104–106), as far as I can make it, only seems to support the weaker claim (Satz 5) that there are multiple analytic structures, not how many there are, nor any relation with a predetermined $C^\infty$ structure (the paper says nothing about $C^\infty$). The other paper cited in the Wikipedia article, by P. J. Nyikos, “Various smoothings of the long line and their tangent bundles”, Advances in Math. 93 (1992) 129–213, doesn't mention analytic structures at all except to note in the introduction that they can exist. So now I am completely confused as to what is known (perhaps as a folklore result with no precise reference). Specifically:
Question: Given $r<s$ in $\{0,1,2,\ldots,\infty,\omega\}$, how many different $C^s$ structures on the long line (or half-line) can exist that are compatible with a given $C^r$ structure? What are the most precise known results in this direction, and, specifically, is it correct that, as Wikipedia claims, for $r=\infty$ and $s=\omega$, every $C^\infty$ structure admits infinitely many different $C^\omega$ structures that are compatible with it? Are there other references than the two papers cited in Wikipedia (citations reproduced above)?
