Suppose we are given $(\mathbb{R}^8,\Phi)$, where $\Phi$ is the self-dual 4-form that defines $Spin(7)\subset SO(8)$ (Cayley calibration, see [Notes on the Octonians](http://arxiv.org/pdf/1005.2820v1.pdf), page 23). Now some 4-subspaces $V$ of $\mathbb{R}^8$ have the property that $\Phi|\_V=\pm vol(V)$ (in general it is $\leqslant$ in absolute value).


**QUESTION 1: Which is the dimension of the set of calibrated subspaces as a submanifold of the Grassmannian $G(4,8)$?**


Now suppose we have a calibrated 8-manifold $(X,\Phi)$.


**QUESTION 2: What can be said about the Cayley submanifold? Is there a Cayley submanifold containing a given Cayley subspace at some point? May be quantified 'how many' Cayley submanifolds are there?**


Any idea is welcome.



Cayley Subspaces in a Calibrated 8-Space
All your questions (and much more) are answered in the 1982 paper *Calibrated Geometries* by Harvey and Lawson. See Acta Mathematica July 1982, Volume 148, Issue 1, pp. 47-157.


Just so you'll know: Harvey and Lawson prove the following 


The dimension of the set $Cay$ of Cayley subspaces in $\mathrm{Gr}(4,8)$ is $12$, in fact, it is a smooth, connected manifold diffeomorphic to $\mathrm{SO}(7)/\left(\mathrm{SO}(3){\times} \mathrm{SO}(4)\right)$. This answers Question 1. Moreover, every $3$-plane in $\mathbb{R}^8$ lies in a unique Cayley $4$-plane.


Every real-anaytic, embedded $3$-manifold $N^3\subset\mathbb{R}^8$ lies in an embedded Cayley $4$-fold $P^4\subset\mathbb{R}^8$. Moreover, any two such Cayley extensions $P\_1$ and $P\_2$ are equal in some open neighborhood of $N^3$. This answers Question 2 for flat space. However, Harvey and Lawson's proof works with essentially no changes for the general case of $(X^8,\Phi)$.


