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I have to show that the direct product of the multi-center Taub-NUT metric with $\mathbb{R}^{3}$ corresponds to a 7-manifold with G2 holonomy.

The metric of the Taub-NUT is:

$ds_{TN}^{2}=V(r)(dr^{2}+r^{2}d\Omega _{2}^{2})+\frac{1}{V(r)}\left ( dy+R\mbox{sin}^{2}\left ( \theta /2 \right )d\phi\right )^{2}$

with $d\Omega _{2}^{2}=d\theta ^{2}+\mbox{sin}^{2}\theta d\phi^{2}$ and $V(r)=1+\frac{R}{2r}$

I don´t know if it will be enough to show that it is Ricci-flat ($R_{mn}=0)$ or trying to find the associative calibration $\Phi$ and show that $\nabla^{g}\Phi=0$ but in this case I don't know how to obtain the components of the three-form $\Phi_{abc}$...

Is there a better way to do so? (I don´t know even if my option would be correct...)

Arctic Char
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Martin Hurtado
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  • There is an equivalent condition to $\nabla^g \phi = 0$, which is $d\phi = 0 = d (\phi)$, where $$ is the Hodge -operator. There should be some relationship between $\phi, \phi$ and the volume form. If you have it, you might be able to compute the two differentials. – Gibbs Nov 07 '17 at 23:41
  • Why do you have to show it? What is the correspondence between metric on 3-d manifolds to metrics on 7-d manifolds? There are known constructions of metrics with $G_2$ holonomy. – Moishe Kohan Nov 08 '17 at 02:17

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