Questions on R. Bryant's paper "Calibrated embeddings in the special Lagrangian and coassociative cases"

Question

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.

1. Bryant defines on page 11 in his 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 implicit function theorem but without any success.

2. 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?

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

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

Best regards Mario

Answer 1

I'm afraid that that article does not do a lot of details in the introductory Section 0, just because more complete explanations were already available in earlier articles of mine. Here are some brief answers. Right now, I don't have the time to write out the explanations in greater detail.

1. This is a consequence of the claim, proved in the reference [Br$_2$], that the condition of being torsion-free is $np$ first order PDE for a section $\sigma:M\to S=F/G$ that defines a $G$-structure (where $n$ is the dimension of the manifold and $p$ is the codimension of $G$ in $\mathrm{SO}(n)$). The hypothesis of strong admissibility implies (indeed, it is equivalent to the condition) that all of these equations are captured by the condition of closure of the differential forms associated to the $G$-structure, which is exactly the condition that an $n$-plane $E$ be an integral element of $\mathcal{I}$.

2. If you unwind the definitions, you will see that the spaces ${\frak{h}}_k$ were defined so as to make this equation true. This computation is best carried out up on $F$, where one can write out the definition of $\hat \alpha$ in a coframing of $F$ given by the structure equations and compute $d\hat\alpha$ explicitly. (When I have more time, maybe I can put in a brief description of this computation.)

3. I didn't claim to give the proof that $\mathrm{SU}(n)\subset\mathrm{SO}(2n)$ is regularly presented for all $n$, I just said that it can be proved. You can get an indication of how the proof goes by looking a little further along in the paper where, on pages 13–14, I do (briefly) give the argument for $n=3$, the case of most interest in the article.