Editing Holonomy (section)

===Special holonomy and spinors===
Manifolds with special holonomy are characterized by the presence  of parallel [[spinor]]s, meaning spinor fields with vanishing covariant derivative.<ref name="Lawson">{{harvnb|Lawson|Michelsohn|1989|loc=§IV.9–10}}</ref>  In particular, the following facts hold:
* Hol(ω) ⊂ ''U''(n) if and only if ''M'' admits a covariantly constant (or ''parallel'') projective pure spinor field.
* If ''M'' is a [[spin structure|spin manifold]], then Hol(ω) ⊂ ''SU''(n) if and only if ''M'' admits at least two linearly independent  parallel pure spinor fields.  In fact, a parallel pure spinor field determines a canonical reduction of the structure group to ''SU''(''n'').
* If ''M'' is a seven-dimensional spin manifold, then ''M'' carries a non-trivial parallel spinor field if and only if the holonomy is contained in G<sub>2</sub>.
* If ''M'' is an eight-dimensional spin manifold, then ''M'' carries a non-trivial parallel spinor field if and only if the holonomy is contained in Spin(7).
<!--Anyone know of nice results specific to the hyper-Kaehler and quaternion-Kaehler holonomies?-->

The unitary and special unitary holonomies are often studied in connection with [[twistor theory]],<ref>{{harvnb|Baum|Friedrich|Grunewald|Kath|1991}}</ref> as well as in the study of [[almost complex structure]]s.<ref name="Lawson" />