[QUESTION]
Let $(S^n,\bar{g})$ be the unit sphere and $h$ be another Riemannian metric on $S^n$, $0<\alpha<1$. $M^{2+\alpha}(S^n):=\left\{F:S^n\stackrel{C^{2,\alpha}}\to S^n\right\}$.
For $F:(S^n,\bar{g})\to(S^n,h)\in M^{2+\alpha}(S^n)$, let
\begin{eqnarray}
X(F):=h_{\circ F}(\Delta F, dF(\cdot))\in A^1(S^n):=\{C^{0,\alpha} 1\text{-forms on }S^n\}
\end{eqnarray}
Then how can we describe the Frechet derivative $DX(F):\ast\stackrel{\text{linear}}\to A^1(S^n)$ of $X:M^{2+\alpha}(S^n)\to (A^1(S^n), \text{the }C^{0,\alpha}\text{-norm})$ at $F\in M^{2+\alpha}(S^n)$ and the implicit function theorem around $F$ ? Can we take a coordinate chart around $F$ as in finite-dimensional cases ?
[BACKGROUND]
If $h=\bar{g}$, $F=Id_{S^n}$, then $X(F)=0$. I want to look for zero points of $X$ around $Id_{S^n}$ when $h$ is close to $\bar{g}$ in some sense. But I am confusing since $M^{2+\alpha}(S^n)$ is not an open set of any Banach spaces.