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Weak Spin(9)-structures on 16-dimensional Riemannian manifolds. (English) Zbl 1021.53028
This paper can be considered as the main source of information about weak $$Spin(9)$$-holonomies which appear in the case when the structure group of the frame bundle reduces to the subgroup $$Spin(9)$$. It was proved by Gray that only three groups which are $$U(n)$$, $$G_2$$, and $$Spin(9)$$ can serve as weak holonomy groups being not the holonomy groups. The latter two cases may appear only in dimensions 7 and 16, respectively. During the last 30 years the first two cases were studied and in the present paper the latter case, which was neglected until now, is considered. The author studies topological properties of manifolds with $$Spin(9)$$-holonomy deriving some conditions and equalities for Pontrjagin and Stiefel-Whitney classes, computes first fourteen homotopy groups of the space $$SO(16)/Spin(9)$$, introduces 16 different classes of weak $$Spin(9)$$-structures, and derives the differential equations satisfied by the unique self-dual 8-forms assigned to weak $$Spin(9)$$-structures of any type. The author also constructs the corresponding twistor spaces studied in details for structures of the so-called nearly parallel type.

##### MSC:
 53C29 Issues of holonomy in differential geometry 53C10 $$G$$-structures 53C28 Twistor methods in differential geometry
##### Keywords:
holonomy of Riemannian manifolds; twistor spaces
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