Restriction of "Spin(7) 4-form" to $\mathbb{R}_+\times S^7$

Question

I asked this question on stackexchange (https://math.stackexchange.com/questions/2212226/restriction-of-spin7-4-form-to-mathbbr-times-s7) but was advised to ask again here:

I'm currently reading through Jason Lotay's paper "Associative Submanifolds of the 7-sphere" (https://arxiv.org/pdf/1006.0361v1.pdf) and the corresponding slides. To construct a positive 3-form on $S^7$ corresponding to a $G_2$-structure he first constructs the "$Spin(7)$ 4-form" $\Phi_0$ on $\mathbb{R}^8=\mathbb{R}\oplus\mathbb{R}^7$ as

$$\Phi_0=dx_0\wedge\phi_0+\star\phi_0$$

where $\phi_0$ is the "associative 3-form on $\mathbb{R}^7$ and $\star$ is the Hodge dual. He then claims that since $\Phi_0$ is self dual, the restriction to $\mathbb{R}^8 - \{0\}=\mathbb{R}_+\times S^7$ is given by

$$\Phi=r^3dr\wedge \phi + r^4\star\phi$$

where $\phi$ is the desired 3-form on $S^7$. However, it is not immediately clear to me how he arrives at this equality.

Answer 1

One way of looking at this is that you know that $\Phi_0$ induces the standard Euclidean metric $g_{0}$ on $\mathbb{R}^8$ and that the restriction of this metric to $S^7$ is the round metric, i.e. $$g_{0}=dr \otimes dr + r^2 g_{S^7}.$$ Moreover the volume forms of $\mathbb{R}^8$ and $S^7$ (with the induced orientation) are related by $$vol_0=r^7 dr \wedge vol_{S^7}.$$ You can define the $G_2$-structure $\phi$ on the unit $S^7$ by $\phi=\iota_{\partial_r} \Phi_0$. The expression for $$\Phi=r^3 dr \wedge \phi + r^4 \star \phi$$ that you give now follows from this and the fact that $\Phi$ is self-dual i.e. $\star_0(r^3 dr \wedge \phi)=r^4 \star \phi$.

Note that the $\star$ appearing here is with respect to the round metric and volume form of $S^7$. However $\Phi$ is a self-dual 4-form with respect to the Hodge star $\star_0$ of the Euclidean metric and volume form of $\mathbb{R}^8$. The expression is essentially obtained by comparing these two Hodge stars and taking into account the scaling factor of r.