Degrees of Maps between Isotropic Grassmann Manifolds

TL;DR

This paper investigates the degrees of maps between isotropic Grassmann manifolds and real Grassmann manifolds, establishing conditions under which these degrees are zero or when the maps are essentially identity.

Contribution

It proves that nontrivial maps between isotropic Grassmannians of the same dimension are only possible when they are essentially the same manifold, otherwise their degree must be zero.

Findings

Degree of maps between isotropic Grassmannians is zero unless they are identical.

Maps between isotropic and real Grassmann manifolds have zero degree under certain dimension conditions.

The results classify possible maps based on their degrees and manifold dimensions.

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.