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 1 Let
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.