I am reading the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by R.Bryant (here the link: http://arxiv.org/abs/math/9912246) and there are certain things that are unclear to me.

Bryant defines on page 11 in is paper the set $V_{n}(\mathcal{I},\pi) =$ {$E\in V_{n}(\mathcal{I})|D_{u}\pi : E \rightarrow T_{\pi(u)}M$ is injective} and claims that $V_{n}(\mathcal{I},\pi) \subset Gr_{n}(TF)$ is a submanifold of codimension $np$, where $p$ is the codimension of $G \subset SO(n)$ in $SO(n)$. My question is: why is this so? I tried to use several chart representations or the implicite function theorem but without any succes.

On page 12 there are the subspaces $\mathfrak{h}_{k}$ defined. Then he computes that $H(E _ {k}) = E + ( \mathfrak{h} _ {k} ) _ {u}$. Why does this hold?

How does he show that $SU(n)$ is reglarly presented in $SO(2n)$?

I hope that some of you have the answers to some of my questions.

Best regards Mario