# Vector cross products on manifolds

##### *2019-10-14 19:37*

Dec 27, 2009 Definition of cross product in Spivak's 'Calculus on Manifolds' Thread (xyyz)2[itex of two quaternions is a pure imaginary quaternion whose imaginary part is equal to the vector cross product of their imaginary parts. This extends to the octonions, and hence to a seven dimensional cross product. Definition of cross product in SpivakVector 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 cross products on manifolds

Further, vector cross products have been studied from an algebraic standpoint in Brown and Grey. Vector cross products are interesting for three reasons: first, they are a natural generalization of the concept of almost complex structure; secondly, a vector cross product on a manifold M generates unusual almost complex structures