Formally integrable complex structures on higher dimensional knot spaces

Let S be a compact oriented finite dimensional manifold and M a finite dimensional Riemannian manifold, let Immf (S, M) the space of all free immersions phi : S -> M and let B-i(,f)+ (S, M) the quotient space Imm(f) (S, M)/Diff(+) (S), where Diff(+) (S) denotes the group of orientation preserving diffeomorphisms of S. In this paper we prove that if M admits a parallel r-fold vector cross product chi is an element of Omega(r) (M, TM) and dim S = r - 1 then B(i)(,f)(+)f (S, M) is a formally Kahler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that S is a codimension 2 submanifold in M, and S = S-1 or M is a torsion-free G(2)-manifold respectively.