Let $M$ be a closed oriented smooth $4$-manifold with $m=b^+_2(M)>0$. We set $\mathcal R$ to be the space of smooth metrics on $M$ and consider the following map $f:\mathcal R \to Gr(m,n)$ defined for any $g \in \mathcal R$, $$ g \to [\alpha_1,\alpha_2,\cdots, \alpha_m]. $$ Here, $\{\alpha_i\}$ is a basis of self-dual harmonic two-forms defined by $g$ and $n=b_2(M)$.
Can we prove the map $f$ is a submersion?
