Immersions in a Quaternionic Grassmannian inducing a given 4-form

Abstract

Let $Gr_k(\H^n)$ be the Grassmannian manifold of Quaternionic $k$-planes in $\H^n$ and let $\gamma^n_k\to Gr_k(\H^n)$ denote the Stiefel bundle of quaternionic $k$-frames in $\H^n$. Let $\sigma$ denote the first symplectic Pontrjagin form associated with the universal connection on $\gamma^n_k$. We show that every 4-form $\omega$ on a smooth manifold $M$ can be induced from $\sigma$ by a smooth immersion $f:M\to Gr_k(\H^n)$ (for sufficiently large $k$ and $n$) provided there exists a continuous map $f_0:M\to Gr_k(\H^n)$ which pulls back the cohomology class of $\sigma$ onto that of $\omega$.