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In Klingenberg's Lectures on Closed Geodesics, before constructing the differentiable structure of the free loop space of a compact manifold $M$, he states that:

A large part of the construction which immediately follows would also work for a non-compact Riemannian manifold. For the finer aspects of the theory, however, compactness is necessary.

Since he gives no references, I ask here: what are the "finer aspects of the theory", and what would not work?

For example, he uses compactness to get a neighbourhood $\mathcal{O}_{\epsilon}$ of the $0$-section in $TM$ that is diffeomorphic to a neighborhood of the diagonal in $M\times M$ by the exponential map, but maybe he could get a neighbourhood not so " nice" and still get the rest of things to work. But he also uses compactness in proving that the resulting loop space has a countable base, and this time it seems to me that using compactness is unavoidable.

These are examples... and they touch only the introduction to the theory, but I hope they help to understand what I'm asking for.

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