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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, 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.

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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 $(X^8,\Phi)$.

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  • $\begingroup$ Thank you very much! Although the whole paper is worth reading, do you remember (more or less) which chapter or section deals with the questions before? $\endgroup$
    – Jjm
    Commented May 6, 2015 at 12:14
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    $\begingroup$ The parts you want are in Chapter IV (The Exceptional Geometries), but I don't remember the exact subsections. They are clearly labeled, though, so you should have no problem finding what you want. $\endgroup$
    – Robert Bryant
    Commented May 6, 2015 at 12:39

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