Almost complex structures on the orthogonal twistor bundle

We give a construction of 2s, s = n(n – 1)/2, many natural almost complex structures on the orthogonal twistor bundle over a 2n-dimensional Riemannian manifold. The usual almost complex structures are then characterised by the condition that they correspond to integrable invariant complex structures on the standard fibre which is identified with the hermitian symmetric space SO(2n)/U(n).