Let us recall that an affine structure on manifold $Y$ (smooth, of dimension $n$) is given by a torsion-free flat connection $\nabla$ on the tangent bundle $TY$.

We will give below three equivalent definitions of the notion of an integral affine structure.

It is easy to see that if $Y$ carries a ${\bf Z}$-affine structure then for any point $y\in Y$ there exist small neighborhood $U$, local coordinate system $(x_{1},...,x_{n})$ in $U$ such that $\nabla=d$ in coordinates $(x_{1},...,x_{n})$, and the lattice $(T_{x}Y)^{{\bf Z}},x\in U$ is a free abelian group generated by the tangent vectors $\partial/\partial x_{i}\in T_{x}Y,1\leq i\leq n$. Let us call ${\bf Z}$-affine such a coordinate system in $U$ (sometimes we will call such $U$ a ${\bf Z}$-affine chart). For a covering of $Y$ by ${\bf Z}$-affine charts the transition functions belong (locally) to $GL(n,{\bf Z})\ltimes{\bf R}^{n}$. Explicitly, a change of coordinates is given by the formula

where $(a_{ij})\in GL(n,{{\bf Z}}),(b_{i})\in{{\bf R}}^{n}$.

Hence, Definition 1 is equivalent to the following

In the above definition $Y$ is just a topological manifold, $C^{\infty}$-structure on it can be reconstructed canonically from ${\bf Z}$-affine structure.

We can restate the notion of ${\bf Z}$-affine structure in the language of sheaves of affine functions.

We say that a real-valued function $f$ on ${{\bf R}}^{n}$ is ${\bf Z}$-affine if it has the form

where $a_{1},\dots,a_{n}\in{\bf Z}$ and $b\in{\bf R}$. We will denote by $Aff_{{\bf Z},{{\bf R}}^{n}}$ the sheaf of functions on ${{\bf R}}^{n}$ which are locally ${\bf Z}$-affine.

Equivalence of the last two definitions follows from the observation that a homeomorphism between two open domains in ${{\bf R}}^{n}$ preserving the sheaf $Aff_{{\bf Z},{{\bf R}}^{n}}$ is given by the same formula $x^{\prime}=A(x)+b,A\in GL(n,{{\bf Z}}),b\in{{\bf R}}^{n}$ as the change of coordinates between two ${\bf Z}$-affine coordinate systems.