  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.
