It is satisfactory to have a nice functional analytic setting for the energy functional in Riemannian geometry. I'm currently deep into Klingenberg's book "Riemannian geometry" which (among other things) aims to construct this Hilbert manifold.
Klingenberg takes as a model Hilbert space the sections $H^1(\gamma^*TM)$ of $H^1$ regularity where $\gamma$ is smooth loop. Then charts are obtained by exponentiating the $H^1$ sections along these loops to reach loops in $M$. One can check that the coordinate changes are smooth. This is also the approach I found in most other sources.
This has the disadvantage that we do not have natural charts around the non-smooth loops, and to check that for example the energy functional is smooth is a bit painful at non-smooth loops.
So why are the charts only defined at smooth loops? An $H^1$ section of $\gamma^*TM$ seems to me well defined for an $H^1$ loop $\gamma$: Namely these are all the maps $\sigma:S^1\rightarrow TM$ that lift $\gamma:S^1\rightarrow M$ and are $H^1$ in all charts in of $TM$. One can still define the covariant derivative along the curve and take the usual $H^1$ norm of such sections using this covariant derivative.
Does something go wrong if we add such charts? Can we not check that the Hilbert manifold is smooth? In the end the proof that the coordinate changes are smooth boils down to the fact that the exponential mapping (on $M$) is smooth and postcomposing $H^1$ loops with a diffeomorphism induces a diffeomorphism on the $H^1$ spaces. I feel like I am missing something important. Can you help me clarify what the problem is (if there is one?)
Can I at least canonically identify $T_\gamma H^1(S^1,M)$ with $H^1(\gamma^*TM)$ for non-smooth loops that I discussed above?
