Flat torus and zero Euler characteristic

Write $q:\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ for the quotient map. For every open interval $I$ of length less than $1$, $q{\mathrm{\mid }}_I$ is injective, and it is open because $q^{-1}(q(U))={\bigcup }_{n\in \mathbb{Z}}(U+n)$ is open for every open $U\subseteq I$. Hence $q{\mathrm{\mid }}_I:I\to q(I)$ is a homeomorphism. Products of these inverses give charts on the product topology of [F2]. In overlapping charts two lifts differ by an integer in each coordinate. This integer pair is locally constant, so each transition is locally an integer translation and is smooth with derivative the identity. The quotient circle is Hausdorff: distinct classes have representatives whose difference has positive distance from $\mathbb{Z}$, and sufficiently short intervals around them have disjoint quotient images. Images of intervals with rational endpoints form a countable basis; their finite products do so on $T^2$. These charts therefore define a Hausdorff second-countable smooth surface without boundary, oriented by their positive Jacobians.

Put the $3\times 3$ grid with vertices $(i\mathrm{/}3,j\mathrm{/}3)$, $0\le i,j\le 3$, on the square $[0,1{]}^2$ and split each of the nine squares by the diagonal from its lower-left to its upper-right corner. The integer-translation identifications $(x,0)\sim (x,1)$ and $(0,y)\sim (1,y)$ map grid cells to grid cells and are injective away from the boundary, so the subdivision descends to a finite curvilinear triangulation of $T^2$: the four interior grid vertices are four classes and the twelve boundary grid vertices form five classes under the two pairings, so $V=4+5=9$; the twelve interior grid edges are distinct classes and the twelve boundary segments form six classes, while the nine diagonals are pairwise distinct, so $E=12+6+9=27$; and the nine squares split into $F=18$ triangles. By [F5], $\chi (T^2)=9-27+18=0$.

Choose quotient charts from products of intervals shorter than one unit in each coordinate. On overlaps their changes are locally integer translations by step 1.1, whose derivatives are the identity, so the local positive tensors $d{x}^2+d{y}^2$ agree and define a smooth metric on $T^2$. Its coordinate frame $({\mathrm{\partial }}_x,{\mathrm{\partial }}_y)$ is orthonormal with constant coefficients, so by [F3] all Christoffel symbols vanish and hence ${\mathrm{\nabla }}_{{\mathrm{\partial }}_x}{\mathrm{\partial }}_x={\mathrm{\nabla }}_{{\mathrm{\partial }}_x}{\mathrm{\partial }}_y={\mathrm{\nabla }}_{{\mathrm{\partial }}_y}{\mathrm{\partial }}_y=0$. With $e_1={\mathrm{\partial }}_x$, $e_2={\mathrm{\partial }}_y$ the connection form is $\omega (X)=g({\mathrm{\nabla }}_X{e}_1,e_2)=0$ for every $X$, so $d\omega =0$; [F4] gives $KdA=0$, and since $dA\mathrm{\ne }0$ pointwise, $K\equiv 0$ on the chart. The quotient charts cover $T^2$, so $K\equiv 0$ and ${\int }_{T^2}KdA=0$.

By step 2.1 the total curvature is $0$, and by step 1.2 $\chi (T^2)=0$; the identity [F1] therefore reads $0=2\pi \cdot 0$, which is true. The torus is compact with empty boundary by [F6], so the global theorem applies.

No new choice is made: the quotient charts, the grid and the diagonal pattern are explicit, and AC licenses the inherited global Gauss-Bonnet theorem [F1] and the countable-choice assumption of the curvature structure equation in [F4].