In this post, we briefly introduce the Lie group , -structures on a manifold and a -manifold. Let us denote the three form on by . We set . This three form is non-degenerate in the sense that whenever we have two linearly independent vectors in , we can find a third vector such that the evaluation of on these vectors is non-zero. We define . One may prove that is a -dimensional Lie subgroup of .

Let us give a different descriptions of . So, it does not look completely arbitrary. is the highest dimension that one may define a cross product. After we identify with the octonions equiped with some octonion product, for any two imaginary octonions we define the cross product to be

Then, we may define the -form on by where the inner product is the standard inner product. Of course, there is a choice on the octonion product and hence, 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 is indeed a -form and then, evaluate it on the basis elements to see how the octonion product should be defined.

Using , we obtain and thus, the above definition is equivalent to

To prove that is alternating, it is enough to prove and as we may replace by to get the desired equalities. However, also note that . Therefore, the first two equalities are enough. It is clear that . Hence, we have the first equality. Furthermore,

Thus, we have showed that is alternating.

Our next goal is to define the octonion product. Clearly, from the explicit definition, we want . In other words, . So, a natural choice for the product is . Similarly, we can choose , , , , and . Of course, as we are describing octonion multiplication, we should also define the multiplication with the th generator but it is the generator of 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 is non-degenerate. Because so, we only need to show that is non-zero for linearly independent and . However, that is a built-in property for a cross-product.

A -structure on a manifold can be defined as a subbundle of the frame bundle of whose fibers are isomorphic to . However, there is an equivalent, more convenient definition. In fact, this definition will follow the scheme of the previous post. More explicitly, since fixes on , we may pull it back to each space to have a three form 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 . So, having a three form on such that for any point , and can be identified by an isomorphism between and , means that we can find a -structure on . By an abuse of notation, we call a manifold with a -structure. Furthermore, since is a subgroup of , it also fixes the standard metric and orientation on giving rise to a Riemannian metric and orientation on the manifold. This immediately implies a non-orientable manifold does not admit a -structure.

Next, we introduce -manifolds. Given a manifold with a -structure and the induced metric , let be the Levi Civita connection on . If , we call a torsion free -structure. A manifold with a torsion free -structure is called a -manifold. In fact, there are a number of ways to define -manifolds, as we can see in the following proposition.

Proposition 1Let be a -structure on with the induced metric and the Levi Civita connection . Then, the following are equivalent:

- and .

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