Vector cross products on manifolds

ALFRED GRAYO1. Introduction.Calabi 6 has shown that every orientable hypersurface of F7 possesses an almost complex structure.The properties of these manifolds have been investigated by Calabi 6, Gray 11, and Yano and Sumitomo 22.The almost complex structure, which is a generalization of the one on Se, is defined in terms of a vector cross product on F7, which is a generalization of the ordinary Gibbs vector cross product on F3.In this paper we give a general definition of vector cross product ( 2), and then study vector cross products on manifolds.Vector cross products are interesting for three reasons : first, they are themselves natural generalizations of the notion of almost complex structure ; secondly, a vector cross product on a manifold M gives rise to unusual almost complex structures on certain submanifolds of M; thirdly, vector cross products provide one approach to the study of Riemannian manifolds with holonomy group G2 or Spin (7).Vector cross products on vector spaces have been studied from an algebraic standpoint in Brown and Gray 5 and from a topological standpoint in Eckmann 9 and Whitehead 20.We consider the topological existence of vector cross products on manifolds in 2.Then in 3 we give conditions for the existence of vector cross products on vector bundles over CW-complexes in terms of characteristic classes.As an extra feature of our investigations we determine the behavior of the triality automorphism of Spin (8) on Spin characteristic classes.In 4 we discuss vector cross products from a differential geometric point of view and in 5 we develop some important relations between vector cross products and curvatures.The rest of the paper ( 6 and 7) is devoted to almost complex structures on manifolds.We show in 6 that every orientable 6-dimensional submanifold of F8 possesses an almost complex structure.Such an almost complex structure is defined by means of a 3-fold vector cross product on F8.We also show that all of Calabi's manifolds can be obtained by this method.However, the almost complex structures constructed here are more general in several ways.In the first place, we consider 6-dimensional orientable submanifolds M of arbitrary 8-dimensional pseudo-Riemannian manifolds M possessing 3-fold vector cross products, e.g., parallelizable 8-dimensional manifolds.Secondly, the metric of M we use may be indefinite provided it has signature (4, 4).(Here we