Almost Complex Structures on Tensor Bundles

Abstract

This chapter investigates a problem for tensor bundles T r s M. If a Riemannian manifold M admits an almost complex structure then so does T r s M provided r + s is odd. If r + s is even a further condition is required on M. The proofs depend on some generalizations of the notions of lifting vector fields and derivations on M, which are defined only for tangent bundles and cotangent bundles. It is shown how vector fields on T r s M can be induced from vector fields, tensor fields of type (r, s), and derivations on M. The main problem—that is, to determine a class of tensor bundles that admit almost complex structures, is considered. For this purpose, it is sufficient to consider contravariant tensor bundles because a Riemannian metric tensor field induces a fibre preserving diffeomorphism of T r s M→T r+S M.