Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\mathbb{R}^n$. Given a manifold $M$ of dimension $m$, are there any methods to study whether $M$ can be embedded into $G_k(\mathbb{R}^n)$ and $AG_k(\mathbb{R}^n)$?
When studying the embedding/immersion problem of manifolds into Euclidean spaces, I learned that there are some obstructions by Stiefel-Whitney class from lecture notes. How about embedding problems into Grassmannians?