Degrees of Maps between Isotropic Grassmann Manifolds

Abstract

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

10 pages