Degrees of Maps between Isotropic Grassmann Manifolds

Abstract

Let $\widetilde{I}_{2n,k}$ denote the space of $k$-dimensional, oriented isotropic subspaces of $\mathbb{R}^{2n}$, called the oriented isotropic Grassmannian. Let $f \colon \widetilde{I}_{2n,k} \rightarrow \widetilde{I}_{2m,l} $ be a map between two oriented isotropic Grassmannians of the same dimension, where $k,l \geq 2$. We show that either $(n,k) = (m,l)$ or the degree of $f$ must be zero. Let $\mathbb{R}\widetilde{G}_{m,l}$ denote the oriented real Grassmann manifold. For $k,l \geq 2$ and $\dim{\widetilde{I}_{2n,k}} = \dim{\mathbb{R}\widetilde{G}_{m,l}}$, we also show that the degree of maps $g \colon \mathbb{R} \widetilde{G}_{m,l} \rightarrow \widetilde{I}_{2n,k} $ and $h \colon \widetilde{I}_{2n,k} \rightarrow \mathbb{R} \widetilde{G}_{m,l}$ must be zero.