Some Observations on projective Stiefel Manifolds

In this note, we compute the upper characteristic rank of the projective Stiefel manifolds over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$ and the flip Stiefel manifolds. We also provide bounds for the $\mathrm{cup}$ lengths of these spaces. We also provide necessary conditions for the existence of $S^3$-map between quaternionic Stiefel manifolds using the Fadell-Husseini index.