Let $(M,g)$ be a 4-dimensional Riemannian manifold and $(P,M,\pi)$ a principal $G$-bundle over $M$. Denote by $ad (P)$ the adjoint bundle of $P$, that is, the vector bundle obtained by dividing $P\times Lie(G)$ by $G$ with the right action $(p,\xi)\mapsto (p\cdot g, Ad(g^{-1})\xi)$.
There is a Hodge star operator for the $ad(P)$-valued 2-forms: $$\star:\Omega^2(M,ad(P))\to\Omega^2(M,ad(P))$$ This is a symmetric isometry and its eigenvalues are $\lambda=\pm 1$. The corresponding eigenspaces will be denoted by $\Omega^2_{\pm}(M,ad(P))$.
The connection 1-forms $\omega$ of $(P,M,\pi)$ whose curvature form $\Omega^\omega$ belongs to the eigenspace $\Omega^2_+(M,ad(P))$ are called self-dual. We will denote by $\tilde{\mathcal M}(P,g)$ the set of self-dual connections.
I am trying to prove the following:
Claim. $\tilde{\mathcal M}(P,g)$ is an infinite dimensional Hilbert submanifold of $\mathcal A(P)$, the (Sobolev complection of) the set of curvature forms
For do so, I propose to use following result:
Subimmersion Theorem (Theorem 3.5.17 of [1]) Let $f:M\to N$ be a smooth map between two Hilbert manifolds and $q\in N$. If $f$ is a subimmersion in an open neigbbourhood of $f^{-1}(q)$. Then $f^{-1}(q)$ is a submanifold of $M$ with $T_pf^{-1}(q)=\ker d_pf$, $p\in f^{-1}(q).
Let us consider the curvature map $$C:\mathcal A(P)\to \Omega^2(M, ad(P)),\,\omega\mapsto C(\omega)=\Omega^\omega$$ Then, if we compose with the projection $\Omega^2(M, ad(P))\to \Omega^2_+(M, ad(P))$ we obtain a map $C^+:\mathcal A(P)\to \Omega^2_+(M, ad(P))$ such that its preimage $(C^+)^{-1}(\{0\})$ coincides with $\mathcal M(P,g)$.
By the definition of subimmersion, it is sufficient to show that $C$ is a submersion. However, this is not the case because its differential $C_{\star \omega}$ coincides with the exterior differential $d_\omega:\Omega^1(M,ad(P))\to \Omega^2(M,ad(P))$, which is not surjective.
Thanks in advance.
References
[1] R. Abraham , J. E. Marsden , T. R. Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences (AMS, volume 75). Third Edition.