Hopf–Rinow on a flat cylinder

By [F1] the circle has lifted charts onto open intervals in $\mathbb{R}$, and [F2] makes their products with real intervals smooth charts for $C$ whose metric matrix is $I_2$. These charts have no half-space boundary, so $C$ is a boundaryless two-dimensional Riemannian manifold. It is nonempty, containing $([0],0)$. Replacing a lift $x$ by $x+m$, $m\in \mathbb{Z}$, changes a lifted coordinate by a translation with identity derivative, so the tangent coordinates $(a,b)$ used in the statement are well-defined.

The cylinder is connected by [F3].

Put $a=x-x^{\mathrm{\prime }}$, let $m=\lfloor a\rfloor$ and $t=a-m\in [0,1)$, and put $k=\lfloor a+1\mathrm{/}2\rfloor$. If $t<1\mathrm{/}2$, uniqueness in [F6] gives $k=m$ and $\delta =k-a=-t$; if $t\ge 1\mathrm{/}2$, it gives $k=m+1$ and $\delta =1-t$. For any integer $j\le m$, one has $\mathrm{\mid }a-j\mathrm{\mid }=a-j\ge t$; for any $j\ge m+1$, one has $\mathrm{\mid }a-j\mathrm{\mid }=j-a\ge 1-t$. Both bounds are attained at $m$ or $m+1$, respectively. Thus $\mathrm{\mid }\delta \mathrm{\mid }=\min \{t,1-t\}\le \mathrm{\mid }a-j\mathrm{\mid }=\mathrm{\mid }{x}^{\mathrm{\prime }}-x+j\mathrm{\mid }(j\in \mathbb{Z}),$ including the tied case $t=1\mathrm{/}2$. If the lifts are changed to $x+r,x^{\mathrm{\prime }}+s$, with $r,s\in \mathbb{Z}$, applying the uniqueness in [F6] to $a+1\mathrm{/}2+r-s$ changes $k$ to $k+r-s$, leaving $\delta$ unchanged. Thus both the nearest displacement and the formula in the statement are independent of the chosen lifts.

For $P=([x],y)$ and $(a,b)\in T_{P}C\cong {\mathbb{R}}^2$, define ${\gamma }_{P,(a,b)}(t)=([x+ta],y+tb),t\in \mathbb{R}\mathrm{.}$ Near each parameter value, a lifted product chart represents this curve by an affine line. The matrix $I_2$ from step 1.1 has zero derivatives, so [F4] gives zero Christoffel symbols and verifies the coordinate geodesic equation. Thus the curve is a geodesic. Its initial data are $P,(a,b)$, and the formula is independent of the representative $x$ by [F1]. Maximal-geodesic uniqueness in [F4] identifies it with the maximal geodesic for those data, whose domain is therefore all of $\mathbb{R}$. Since the initial data were arbitrary, $C$ is geodesically complete, and the time-one definition in [F4] gives ${\exp }_P(a,b)=([x+a],y+b)\mathrm{.}$

Steps 1.1--2.1 verify the nonempty, connected, boundaryless, and geodesically complete hypotheses of [F5]. Hopf--Rinow therefore makes $(C,d_C)$ a complete metric space, makes every closed bounded subset of it compact, and supplies a minimizing geodesic between any two points. This is the asserted completeness, properness, and existence claim.

The curve $\sigma$ in the statement is the restriction of the all-real geodesic in step 2.1 with initial velocity $(\delta ,\mathrm{\Delta }y)$. Its endpoint is $([x+\delta ],y+\mathrm{\Delta }y)=([x^{\mathrm{\prime }}+k],y^{\mathrm{\prime }})=Q$ by [F1]. Its speed is the constant $\sqrt{{\delta }^2+(\mathrm{\Delta }y{)}^2}$, so [F7] gives the same number for its length.

The exponential formula in step 2.1 gives ${\exp }_P(0,0)=P={\exp }_P(1,0)\mathrm{.}$ The two tangent vectors are distinct, so every fibre exponential map is noninjective.

Under [A1], let $\eta :[0,1]\to C$ be the minimizing geodesic from $P$ to $Q$ supplied in step 3.1. By step 2.1 it has the form $\eta (t)=([x+tA],y+tB)$ for its initial velocity $(A,B)$. The endpoint condition and [F1] give $B=\mathrm{\Delta }y$ and $A=x^{\mathrm{\prime }}-x+j$ for some $j\in \mathbb{Z}$. Therefore [F2], [F7], and step 1.3 give $L(\eta )=\sqrt{(x^{\mathrm{\prime }}-x+j{)}^2+(\mathrm{\Delta }y{)}^2}\ge \sqrt{{\delta }^2+(\mathrm{\Delta }y{)}^2}=L(\sigma )\mathrm{.}$ But $L(\eta )=d_C(P,Q)$, while the infimum definition [F7] gives $d_C(P,Q)\le L(\sigma )$. Equality holds throughout. Thus $\sigma$ is minimizing and the displayed distance formula is proved.

If $P=Q$, the fibre criterion in [F1] says $x-x^{\mathrm{\prime }}$ is an integer, so the uniqueness in [F6] gives $k=x-x^{\mathrm{\prime }}$, $\delta =0$, and $\mathrm{\Delta }y=0$; thus $\sigma$ is the constant zero-length geodesic. When $\mathrm{\mid }\delta \mathrm{\mid }=1\mathrm{/}2$, the adjacent integer translate gives a second minimizer of the same length; uniqueness is not claimed. The closed parameter endpoints $0,1$ were evaluated in step 3.2. The cylinder is explicitly nonempty and two-dimensional, while the one-dimensional periodic factor and the period-one tangent vector are exactly what produce step 3.3; no empty or zero-dimensional case is being asserted. There is no iff claim in this example. Assumption [A1] is used only through [F4]--[F5] in steps 2.1, 3.1, and 4.1; the explicit formulas make no choices.