Vector cross products

Vector Cross Products on Manifolds

“…It is easily verified that P2 is a 3-fold vector cross product on M. Furthermore, F2 is a vector cross product of the opposite kind to Px, because [5] on each tangent space of M, P and F2 determine different kinds of vector cross products.…”

“…However, he assumed they were continuous instead of multilinear and used only the positive definite bilinear form. In the continuous case vector cross products are classified in [9] and [20] ; for the multilinear case (over an arbitrary field) the classification is given in [5].…”

“…Let IF be the orthogonal complement of a and b in V. Define J: W'-> W by Jx=P(a, b, x), for xe W. Then [5] J is an almost complex structure on W (see also Theorem (2.6)). The group Ax is a connected Lie group and its Lie algebra 91,. is isomorphic to the derivations of F (i.e., maps D:V^-V such that DP(x, y, z) = P(Dx, y, z)+P(x, Dy, z) + P(x,y, Dz)) which annihilate a and b.…”

“…The proof of (iv) is the same as that of (iii) with F7 replaced by F8. 5. Relations between curvature and vector cross products.…”

“…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].…”