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In this post, we briefly introduce the Lie group ${G_2}$ , ${G_2}$ -structures on a manifold and a ${G_2}$ -manifold. Let us denote the three form ${dx^i\wedge dx^j \wedge dx^k}$ on ${{\mathbb R}^7}$ by ${dx^{ijk}}$ . We set ${\phi_0 = dx^{123}+ dx^{145}-dx^{167}+dx^{246}+dx^{257}+dx^{347}-dx^{356}}$ . This three form is non-degenerate in the sense that whenever we have two linearly independent vectors in ${{\mathbb R}^7}$ , we can find a third vector such that the evaluation of ${\phi_0}$ on these vectors is non-zero. We define ${G_2 = \left\{ M\in GL(7,{\mathbb R}) \big| M^*\phi_0 = \phi_0 \right\}}$ . One may prove that ${G_2}$ is a ${14}$ -dimensional Lie subgroup of ${SO(7)}$ .

Let us give a different descriptions of ${\phi_0}$ . So, it does not look completely arbitrary. ${7}$ is the highest dimension that one may define a cross product. After we identify ${{\mathbb R}^8}$ with the octonions ${\mathbb O}$ equiped with some octonion product, for any two imaginary octonions ${x,y \in {\mathbb R}^7 \cong im(\mathbb O)}$ we define the cross product to be

$\displaystyle \begin{array}{rcl} x \times y = \frac{1}{2} [x,y] = \frac{1}{2}(xy-yx). \end{array}$

Then, we may define the ${3}$ -form on ${{\mathbb R}^7}$ by ${\phi_0(x,y,z) = \left< x \times y, z\right>}$ where the inner product is the standard inner product. Of course, there is a choice on the octonion product and hence, ${\phi_0}$ may be different than the one we explicitly wrote above. However, we show that they are equivalent using the right octonion product. To show they are equivalent; first, we prove that ${\left<x\times y, z\right>}$ is indeed a ${3}$ -form and then, evaluate it on the basis elements to see how the octonion product should be defined.

Using ${im(xy)=-im(yx)}$ , we obtain ${x\times y = im(xy)}$ and thus, the above definition is equivalent to

$\displaystyle \begin{array}{rcl} \phi_0(x,y,z) &=& \left< xy, z\right>. \end{array}$

To prove that ${\phi_0}$ is alternating, it is enough to prove ${\phi_0(x,x,y)=0, \phi_0(x,y,x)=0}$ and ${\phi_0(y,x,x)=0}$ as we may replace ${x}$ by ${x+z}$ to get the desired equalities. However, also note that ${x \times y = - y\times x}$ . Therefore, the first two equalities are enough. It is clear that ${x\times x = 0}$ . Hence, we have the first equality. Furthermore,

$\displaystyle \begin{array}{rcl} \phi_0(x,y,x) &=& \left< xy, x\right> \\ &=& |x|^2\left< y,1\right> \\ &=& 0. \end{array}$

Thus, we have showed that ${\phi_0}$ is alternating.

Our next goal is to define the octonion product. Clearly, from the explicit definition, we want ${\phi_0(x_1,x_2,x_3)=1}$ . In other words, ${\left<x_1 x_2, x_3\right> = 1}$ . So, a natural choice for the product ${x_1x_2}$ is ${x_3}$ . Similarly, we can choose ${x_1x_4=x_5}$ , ${x_1x_6=-x_7}$ , ${x_2x_4=x_6}$ , ${x_2x_5=x_7}$ , ${x_3x_4=x_7}$ and ${x_3x_5=-x_6}$ . Of course, as we are describing octonion multiplication, we should also define the multiplication with the ${8}$ th generator but it is the generator of ${Re(\mathbb O)={\mathbb R}}$ part. So, it is just the trivial multiplication i.e. the multiplication coming from the vector space structure. We do not show that this indeed defines an octonion product.

Next, we need to show that they are equal and to do that, it is enough to evaluate on the basis elements. It is an easy computation which we omit.

Note that this definition makes an earlier claim more plausible, namely that ${\phi_0}$ is non-degenerate. Because ${\phi_0(x,y,x\times y) = \left<x\times y,x\times y\right>}$ so, we only need to show that ${x\times y}$ is non-zero for linearly independent ${x}$ and ${y}$ . However, that is a built-in property for a cross-product.

A ${G_2}$ -structure on a manifold ${M}$ can be defined as a subbundle of the frame bundle of ${M}$ whose fibers are isomorphic to ${G_2}$ . However, there is an equivalent, more convenient definition. In fact, this definition will follow the scheme of the previous post. More explicitly, since ${G_2}$ fixes ${\phi_0}$ on ${{\mathbb R}^7}$ , we may pull it back to each space ${T_pM}$ to have a three form ${\phi}$ on the manifold and similarly, if we have such a three form on the manifold, then we may find a subbundle of the frame bundle whose fibers are ${G_2}$ . So, having a three form ${\phi}$ on ${M}$ such that for any point ${p\in M}$ , ${\phi_p}$ and ${\phi_0}$ can be identified by an isomorphism between ${{\mathbb R}^n}$ and ${T_pM}$ , means that we can find a ${G_2}$ -structure on ${M}$ . By an abuse of notation, we call ${(M,\phi)}$ a manifold with a ${G_2}$ -structure. Furthermore, since ${G_2}$ is a subgroup of ${SO(7)}$ , it also fixes the standard metric and orientation on ${{\mathbb R}^7}$ giving rise to a Riemannian metric and orientation on the manifold. This immediately implies a non-orientable manifold does not admit a ${G_2}$ -structure.

Next, we introduce ${G_2}$ -manifolds. Given a manifold ${M}$ with a ${G_2}$ -structure ${\phi}$ and the induced metric ${g}$ , let ${\nabla}$ be the Levi Civita connection on ${(M,g)}$ . If ${\nabla \phi =0}$ , we call ${\phi}$ a torsion free ${G_2}$ -structure. A manifold with a torsion free ${G_2}$ -structure is called a ${G_2}$ -manifold. In fact, there are a number of ways to define ${G_2}$ -manifolds, as we can see in the following proposition.

Proposition 1 Let ${(M^7,\phi)}$ be a ${G_2}$ -structure on ${M}$ with the induced metric ${g}$ and the Levi Civita connection ${\nabla}$ . Then, the following are equivalent:

${\nabla \phi = 0}$

${Hol(g) \subseteq G_2}$

${d\phi = 0}$ and ${d^*\phi = 0}$ .

If any one of the conditions of the proposition holds (and hence, all), we call ${M}$ a ${G_2}$ -manifold. The first example of a metric with ${G_2}$ holonomy is given by Bryant. The metric in his example is incomplete. Later, Bryant and Salamon constructed complete metrics with ${G_2}$ holonomy on non-compact manifolds. Then, Joyce constructed complete examples on compact manifolds.

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