In mathematics, there are two distinct meanings of the term affine Grassmannian.
Given a finite-dimensional vector space V and a non-negative integer k, then Graffk(V) is the topological space of all affine k-dimensional subspaces of V. It has a natural projection p:Graffk(V) → Grk(V), the Grassmannian of all linear k-dimensional subspaces of V by defining p(U) to be the translation of U to a subspace through the origin.
As a homogeneous space, the affine Grassmannian of an n-dimensional vector space V can be identified with where E(n) is the Euclidean group of Rn and O(m) is the orthogonal group on Rm.
Indeed, a k-dimensional affine subspace of Rn is the locus of solutions of a rank n − k system of affine equations These determine a rank n−k system of linear equations on Rn+1 whose solution is a (k + 1)-plane that, when intersected with xn+1 = 1, is the original k-plane.
Because of this identification, Graff(k,n) is a Zariski open set in Gr(k + 1, n + 1).