Embeddings of Affine Grassmann Spaces

In this paper we prove that if a Grassmann space \( \Delta = Gr_{{\mathbb{A}}}(m, h, K)\) of the h–subspaces of an affine space \( {\mathbb{A}} = AG(m,K)\) has an embedding e into a projective space \(PG(n,K^\prime)\) over a skew–field \(K^\prime\), and e satisfies two suitable conditions (α) and (β), then K and \(K^\prime\) are isomorphic fields and Δ e is, up to projections, an affine Grassmannian.