Let $B$ be a topological $n$-dimensional manifold.

• (i)

  An affine manifold is a pair $(B,\mathscr{A})$ where $B$ is an $n$-dimensional manifold and $\mathscr{A}$ is a maximal atlas on $B$ whose transition maps are $\aff(\mathbb{R})$ transformations. We call $\mathscr{A}$ an affine structure on $B$.

• (ii)

  An affine manifold $(B,\mathscr{A})$ is integral if the transition maps of the affine structure $\mathscr{A}$ are $\aff_{\mathbb{R}}(\mathbb{Z})$ transformations. We call $\mathscr{A}$ an integral affine structure on $B$.

• (iii)

  A continuous map $\alpha:B\rightarrow B^{\prime}$ is (integral) affine if on each local coordinate chart, $\alpha$ is an element of ($\aff_{\mathbb{R}}(\mathbb{Z}^{n},\mathbb{Z}^{n^{\prime}})$) $\aff(\mathbb{R}^{n},\mathbb{R}^{n^{\prime}})$. Two (integral) affine manifolds $B$ and $B^{\prime}$ are said to be (integral) affine isomorphic if there is an (integral) affine homeomorphism between them.