The Bruhat decomposition of real Grassmann manifolds

We study the Grassmann manifold $G_k$ of all $k$-dimensional subspaces of ${\mathbb R}^n$. The Cartan embedding $G_k\subset O(n)$ realizes $G_k$ as a subspace of $Sl_n({\mathbb R})$ and we study the decomposition $G_k=\coprod_w (BwB\cap G_k)$ inherited from the classical Bruhat decomposition. We prove that this defines a CW structure on $G_k$ and determine the incidence numbers between cells.