Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form

Abstract

Let Gr k (H n ) be the Grassmannian manifold of Quaternionic k-planes in H n and let γ n k → Gr k (H n ) denote the Stiefel bundle of quaternionic k-frames in H n . Let σ denote the first symplectic Pontrjagin form associated with the universal connection on γ n k . We show that every 4-form ω on a smooth manifold M can be induced from σ by a smooth immersion f : M → Gr k (H n ) (for sufficiently large k and n) provided there exists a continuous map f 0 : M → Gr k (H n ) which pulls back the cohomology class of σ onto that of ω.