Flat torus model geometry

The quotient charts make $T^n$ a smooth $n$-manifold. For $u\in {\mathbb{R}}^n$ put $B_u:=u+(-1\mathrm{/}2,1\mathrm{/}2{)}^n$. The map $q$ is open: for open $W\subseteq {\mathbb{R}}^n$ one has $q^{-1}(q(W))={\bigcup }_{m\in {\mathbb{Z}}^n}(W+m)$, which is open, so $q(W)$ is open by [F1]. The restriction $q{\mathrm{\mid }}_{B_u}$ is injective: if $q(x)=q(y)$ with $x,y\in B_u$, then $x-y\in {\mathbb{Z}}^n$, while $\mathrm{\mid }{x}^i-y^i\mathrm{\mid }<1$ for every $i$, so $x=y$. An injective continuous open map is a homeomorphism onto its image, so $q(B_u)$ is an open set homeomorphic to the box $B_u$. If $x\in B_u$ and $y\in B_v$ satisfy $q(x)=q(y)$, then $x-y\in {\mathbb{Z}}^n$. For a fixed point of an overlap, write its two unique chart lifts as $x_0\in B_u$ and $y_0\in B_v$, and put $m=y_0-x_0\in {\mathbb{Z}}^n$. In the coordinates of $B_u$, the chart transition is $z\mapsto {\phi }_v(q(z))$; the difference ${\phi }_v(q(z))-z$ is a continuous function on the overlap with values in the discrete set ${\mathbb{Z}}^n$. It is therefore constant on a neighbourhood of $x_0$, where it equals $m$. Thus the transition is locally the translation $z\mapsto z+m$ at every overlap point, hence is smooth; the translation may differ on different connected pieces of the overlap. The charts $q(B_u)$ for all $u\in {\mathbb{R}}^n$ cover $T^n$, since every class $[x]$ has the representative $x\in B_x$, and any two of them are smoothly compatible. They therefore form a smooth atlas. Their domains are homeomorphic to open boxes, so every point has a Euclidean neighbourhood.

$T^n$ is a topological manifold, connected, and $q$ is a surjection. The map $q$ is surjective by definition of the quotient. It is Hausdorff: if $[x]\mathrm{\ne }[y]$, only finitely many $m\in {\mathbb{Z}}^n$ satisfy $\mathrm{\mid }x-y-m\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }+1$, and none of them satisfies $x-y-m=0$, so $\delta :=\min \{\mathrm{\mid }x-y-m\mathrm{\mid }:m\in {\mathbb{Z}}^n\}>0$ (the minimum over the finitely many small displacements is attained, and the remaining displacements are larger than $\mathrm{\mid }x-y\mathrm{\mid }+1$); the open sets $q(U)$ and $q(V)$ of step 1.1 for the balls $U,V$ of radius $\delta \mathrm{/}3$ about $x$ and $y$ are disjoint, because a common point would give $u\in U$, $w\in V$ with $u-w\in {\mathbb{Z}}^n$ and hence $\mathrm{\mid }x-y-(u-w)\mathrm{\mid }\le 2\delta \mathrm{/}3<\delta$. It is second countable: the images under $q$ of the boxes with rational corners form a countable family of open sets, and for open $W\subseteq T^n$ and $[x]\in W$ there is such a box $B$ with $x\in B\subseteq q^{-1}(W)$, so $[x]\in q(B)\subseteq W$. Together with step 1.1 this makes $T^n$ a topological $n$-manifold by [F1]. Finally ${\mathbb{R}}^n$ is path connected, hence connected, by [F6], and $q$ is continuous and surjective, so $T^n$ is connected by [F6].

The flat metric descends, and $q$ is a local isometry. [F1, F2, step 1.1] Define a symmetric $(0,2)$-tensor field $g$ on $T^n$ chartwise: on the chart $q(B_u)$ with inverse chart $\phi =(q{\mathrm{\mid }}_{B_u}{)}^{-1}$, set $g_y(\xi ,\eta ):=g_{\mathrm{E}}(d{\phi }_y(\xi ),d{\phi }_y(\eta )),y\in q(B_u),\xi ,\eta \in T_y{T}^n\mathrm{.}$ On an overlap $q(B_u)\cap q(B_v)$ the two inverses differ by the integer translation of step 1.1, whose differential is the identity of ${\mathbb{R}}^n$ and which preserves the Euclidean metric; hence the two definitions agree on the overlap and glue to a well-defined smooth tensor field. In every chart the matrix of $g$ is the identity, so $g$ is symmetric and positive definite and is a Riemannian metric by [F2]. Since each inverse chart is a local inverse of $q$ composed with a translation, the differential of $q$ is a linear isometry from $({\mathbb{R}}^n,g_{\mathrm{E}})$ onto $(T_{[x]}{T}^n,g_{[x]})$ at every point. The quotient charts also make $q$ a local diffeomorphism, so $q$ is a local isometry and $q^{\ast }g={\sum }_{i}d{x}^i\otimes d{x}^i$. Any metric $g^{\mathrm{\prime }}$ with $q^{\ast }{g}^{\mathrm{\prime }}=g_{\mathrm{E}}$ equals $g$ because $d{q}_x$ is surjective for every $x\in {\mathbb{R}}^n$ and the equality $g_{[x]}^{\mathrm{\prime }}(d{q}_{x}u,d{q}_{x}w)=g_{\mathrm{E}}(u,w)=g_{[x]}(d{q}_{x}u,d{q}_{x}w)$ determines $g^{\mathrm{\prime }}$ on all pairs of tangent vectors.

Completeness and geodesics. [F3, F4, step 2.2] Let $[x]\in T^n$ and $v\in T_{[x]}{T}^n$. Since $d{q}_x$ is surjective, choose $v_0\in {\mathbb{R}}^n$ with $d{q}_x(v_0)=v$. By [F3] the straight line $\sigma (t):=x+t{v}_0$ is the maximal geodesic of ${\mathbb{R}}^n$ with $\sigma (0)=x$, ${\sigma }^{\mathrm{\prime }}(0)=v_0$, and it is defined on all of $\mathbb{R}$. A local isometry sends geodesics to geodesics by [F3], so $\gamma (t):=q(\sigma (t))=[x+t{v}_0]$ is a geodesic of $(T^n,g)$ defined on all of $\mathbb{R}$, with $\gamma (0)=[x]$ and ${\gamma }^{\mathrm{\prime }}(0)=d{q}_x(v_0)=v$. By the uniqueness clause of [F3] it is the maximal geodesic of $(T^n,g)$ with initial datum $([x],v)$. Hence every maximal geodesic of $T^n$ is defined on all of $\mathbb{R}$, so $(T^n,g)$ is geodesically complete, and by Hopf–Rinow [F4] it is a complete metric space.

The torus is flat. [F5, step 2.2] In every quotient chart of step 1.1 the matrix of $g$ is the constant identity matrix. Its first derivatives vanish, so the Christoffel symbols of the chart vanish by [F5], and the coordinate curvature formula of [F5] gives $R=0$ at every point of the chart. The charts cover $T^n$, so the Riemann curvature four-tensor vanishes identically; by [F5] every sectional curvature $K$, being $\mathrm{Rm}(u,v,v,u)$ divided by the positive Gram determinant of an independent pair, vanishes. In dimension $n=1$ there is no tangent two-plane and the curvature assertion is vacuous, as it must be.

The exponential map, its formula and its failure of injectivity. [F3, step 3.1] The quotient chart $\phi =(q{\mathrm{\mid }}_{B_u}{)}^{-1}$ identifies $T_{[x]}{T}^n$ with ${\mathbb{R}}^n$ by its differential $d{\phi }_{[x]}$; under this identification the vector $v\in T_{[x]}{T}^n$ corresponds to $v_0:=d{\phi }_{[x]}(v)\in {\mathbb{R}}^n$, and $d{q}_x(v_0)=v$. Step 3.1 shows that the maximal geodesic with initial datum $([x],v)$ is $t\mapsto q(x+t{v}_0)=[x+t{v}_0]$, defined on all of $\mathbb{R}$; evaluating at $t=1$ gives ${\exp }_{[x]}(v)={\gamma }_v(1)=[x+v_0],$ which in the coordinates of the chart is the formula ${\exp }_{[x]}(v)=[x+v]$ recorded in the Statement. In particular the domain of ${\exp }_{[x]}$ is all of $T_{[x]}{T}^n$. Noninjectivity: $e_1\in {\mathbb{Z}}^n$ is nonzero because $n\ge 1$, and $q(0)=[0]$ while $q(e_1)=[e_1]=[0]$, so ${\exp }_{[x]}(0)=[x]={\exp }_{[x]}(e_1)$ with $0\mathrm{\ne }{e}_1$; hence the exponential map of the flat torus is not injective at any point.

Boundary cases and choice. [A1, F1, step 2.1, step 4.1] The case $n=1$ is the circle $\mathbb{R}\mathrm{/}\mathbb{Z}$: the quotient charts are intervals, and for every initial vector $v\in T_{[x]}{T}^1\cong \mathbb{R}$ the geodesic is the constant-speed winding $t\mapsto [x+tv]$ on a circle of circumference one, with speed $\mathrm{\mid }v\mathrm{\mid }$; the paths $t\mapsto [x\pm t]$ are the unit-speed cases. The failure of injectivity is again produced by the nonzero lattice vector $e_1$. The degenerate velocity $v=0$ gives the constant geodesic through $[x]$ by step 3.1, and it is excluded as a witness of noninjectivity because $e_1\mathrm{\ne }0$; the zero divisor in the metric is absent because the Euclidean metric is positive definite. The lattice ${\mathbb{Z}}^n$ and its cosets are explicit, and no accumulation or limiting argument occurs; the only choice is the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of [A1], used through the maximal-geodesic theorem [F3] and Hopf–Rinow [F4] in step 3.1.