ALMOST COMPLEX STRUCTURES ON THE TENSOR BUNDLES

Let M be an n-dimensional differentiable manifold of class C(infinity), T(q)(p)(M) its tensor bundles of type (p,q). It is well known that the tensor bundle of type (1, q) of M admits an almost complex structure on the pure cross-section, if M admits an almost complex structure. The main purpose of this paper is to investigate a similar problem for tensor bundles T(q)(p)(M) of type (p,q), p > 1. We prove that if a manifold M admits an almost complex structure phi, then so does T(q)(p)(M), p > I on the pure cross-section provided phi is integrable. The proofs depend on some generalizations of the notions of lifting derivations.