Hallo,

I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some answers. The paper can be found here: http://arxiv.org/abs/math/9912246 . Here is my question: On page 12, in the middle, there are defined the spaces $\mathfrak{h}_{k}$ = { $x\in \mathbb{R}^{n\times n}$ | $\iota^{*}_{k}(x.\alpha)=0$, $\forall \alpha \in $ $\Lambda^{*}(\mathbb{R}^{n})^{G}$ }. I am not very familiar with the notation. I think $x.\alpha$ means the Lie-derivative of $\alpha$ in the direction $x$. Am I right? If so, using the formula for the Lie-derivative $\mathcal{L}_{X}= \iota d + d\iota$. I do not see how to plug in a matrix $x \in \mathbb{R}^{n\times n}$ in a form defined on $\mathbb{R}^{n}$. How can this be understood? I think, by using the $\mathbb{R}^{n}$-valued form $\nu$ defined by $\nu (v) = u(\pi ' (v))$ , for all $v \in T _{u} F$ (also on page 12 in the middle). But still, for $x \in \mathbb{R}^{n\times n} = Lie(GL(n,\mathbb{R}))=ker(\pi ')$ one gets $ \nu(x)=0 $. How is this to be understood? I would be very tankfull if somebody could help me with this.

greetings hapchiu

notnecessary to split up formulas in the way you did for the first. It is true there is an issue with displaying braces { , however there is another way around this: namely use two backslashes instead of one so \\{ or use the commands \lbrace and \rbrace . Moreover, the symbols _ and * cause problems as the have an additional meaning here. The latter can be avoided using \ast instead. If issues persist include the problematic formula in backticks ` (both at end and start). – user9072 Feb 20 '13 at 13:47vector fieldon $\mathbb{R}^n$, one whose flow is the $1$-parameter group of linear transformations $e^{tx}$ of $\mathbb{R}^n$. Then $x\ldot\alpha$ is just the Lie derivative of $\alpha$ with respect to $x$ (thought of as a vector field). – Robert Bryant Feb 20 '13 at 15:33