The affine Grassmannian $ \text{Graff} $ is a vector bundle over the (regular/linear) Grassmannian $ \text{Gr} $
$$
\mathbb{R}^k \to \text{Graff}_{n-k}(\mathbb{R}^n) \to \text{Gr}_{n-k}(\mathbb{R}^n)
$$
where $ \text{Graff}_{n-k}(\mathbb{R}^n) $ is the manifold of all
$ n-k $ dimensional affine subspaces of $ \mathbb{R}^n $. And the map $ \text{Graff}_{n-k}(\mathbb{R}^n) \to \text{Gr}_{n-k}(\mathbb{R}^n) $ is given by sending an affine $ n-k $ dimensional subspace to the unique $ n-k $ dimensional linear subspace which is parallel to it.
Comparing to the (regular/linear) Grassmannian, note that while taking orthogonal complements gives a canonical isomorphism $ \text{Gr}_{n-k} \cong \text{Gr}_k $ this is not quite true for $ \text{Graff}_{n-k} $ and $ \text{Graff}_{k} $ since they are vector bundles of different dimensions, $ n-k $ versus $ k $, over the same base space $ \text{Gr}_{n-k} \cong \text{Gr}_k $ (the affine Grassmannians only coincide when $ n-k=k $). indeed, using this canonical isomorphism for the Grassmannian we have
$$
\mathbb{R}^k \to \text{Graff}_{n-k}(\mathbb{R}^n) \to \text{Gr}_{k}(\mathbb{R}^n)
$$
A familiar example of this is $ \text{Graff}_1(\mathbb{R}^2) $
$$
\mathbb{R} \to \text{Graff}_1(\mathbb{R}^2) \to \mathbb{R}P^1
$$
which is just the Moebius strip viewed as the manifold of all lines in the plane (these are affine lines so they don't need to go through the origin). The Moebius strip is known to be the total space of the tautological bundle over $ \mathbb{R}P^1 $. That leads me to ask:
Is $ \text{Graff}_{n-k}(\mathbb{R}^n) $ diffeomorphic to the total space of the tautological bundle of $ \text{Gr}_k(\mathbb{R}^n)$?
Best Answer
Yes, and the diffeomorphism is not difficult to construct. Given an affine $(n-k)$-plane $A\in\operatorname{Graff}_{n-k}(\mathbb{R}^n)$, there is a unique linear $k$-plane $A^\perp\in\operatorname{Gr}_k(\mathbb{R}^n)$ orthogonal to $A$; these intersect at a single point $p$. The map $A\mapsto (A^\perp,p)$ is a diffeomorphism (and in fact a vector bundle isomorphism over $\operatorname{Gr}_{k}(\mathbb{R}^n)$).