A complete manifold with zero global injectivity radius

Periodic coordinate changes have the form $\theta \mapsto \theta +k$ and derivative one, so the displayed tensor is globally well defined. In every product chart its matrix is $\mathrm{diag}(4{\pi }^2{e}^{-2u},1)$, which is smooth, symmetric, and positive definite by [F1]--[F5]. Thus $M$ is a nonempty boundaryless Riemannian two-manifold. Moreover, with $x=2\pi \theta$ and $y=e^u$, one has $\frac{d{x}^2+d{y}^2}{y^2}=4{\pi }^2{e}^{-2u}d{\theta }^2+d{u}^2\mathrm{.}$ Hence this metric is the quotient of the upper-half-plane hyperbolic metric by the translation $x\mapsto x+2\pi$, so it is the complete-cusp candidate asserted in the statement; completeness itself is proved below, not inferred from the quotient picture.

The circle is connected by [F6] and [F7], the real line is connected by [F8], and their product $M$ is connected by [F9].

Let $\gamma :(a,b)\to M$ be any maximal geodesic, supplied by [F10] and [F11]. The periodic coordinate vector ${\mathrm{\partial }}_{\theta }$ and ${\mathrm{\partial }}_u$ form a global frame, because the quotient-coordinate transitions are translations. Write ${\gamma }^{\mathrm{\prime }}(t)=\alpha (t){\mathrm{\partial }}_{\theta }+\beta (t){\mathrm{\partial }}_u,\beta (t)=u^{\mathrm{\prime }}(t)\mathrm{.}$ By [F13] its speed is a constant $c\ge 0$, and therefore $c^2=4{\pi }^2{e}^{-2u(t)}\alpha (t{)}^2+\beta (t{)}^2\mathrm{.}$ In particular $\mathrm{\mid }{u}^{\mathrm{\prime }}(t)\mathrm{\mid }\le c$ and $\mathrm{\mid }\alpha (t)\mathrm{\mid }\le c{e}^{u(t)}\mathrm{/}(2\pi )$.

Fix $u\in \mathbb{R}$ and let $p=p_u$. The based loop ${\mathrm{\ell }}_u(s)=([s],u),0\le s\le 1,$ has constant speed $2\pi e^{-u}$ and hence length $L_u=2\pi e^{-u}$ by [F19]. Its projection to the first factor is the standard degree-one loop. If ${\mathrm{\ell }}_u$ were based-null-homotopic in $M$, composing such a homotopy with that projection would make the degree-one loop null-homotopic in $\mathbb{R}\mathrm{/}\mathbb{Z}$, contrary to [F23]. Thus ${\mathrm{\ell }}_u$ is not null-homotopic.

Suppose $b<+\mathrm{\infty }$ and fix $t_0\in (a,b)$. Put $D=b-t_0$, $A=u(t_0)-cD$, $B=u(t_0)+cD$, and $V=c{e}^B\mathrm{/}(2\pi )$. The mean-value theorem [F14] and step 1.3 give $A\le u(t)\le B,\mathrm{\mid }\alpha (t)\mathrm{\mid }\le V,\mathrm{\mid }\beta (t)\mathrm{\mid }\le c(t_0<t<b)\mathrm{.}$ The closed box $Q=[0,1]\times [A,B]\times [-V,V]\times [-c,c]$ is compact by [F15]. The map $\mathrm{\Psi }(s,r,\xi ,\eta )=\xi {\mathrm{\partial }}_{\theta }{\mathrm{\mid }}_{([s],r)}+\eta {\mathrm{\partial }}_u{\mathrm{\mid }}_{([s],r)}$ from $Q$ to $TM$ is smooth and hence continuous, so $K=\mathrm{\Psi }[Q]$ is compact by [F16]. The displayed bounds put the velocity lift $(\gamma (t),{\gamma }^{\mathrm{\prime }}(t))$ in $K$ for every $t_0<t<b$. The right-endpoint clause of [F17] extends $\gamma$ past $b$, contradicting maximality. Thus $b=+\mathrm{\infty }$.

For $q={\mathrm{\ell }}_u(s)$, the forward subarc has length $s{L}_u$, while the reverse of the subarc from $s$ to $1$ has length $(1-s)L_u$. The length and distance definitions [F19] and [F20] therefore give $d_g(p,q)\le L_u\min \{s,1-s\}\le \frac{L_u}{2}=\pi e^{-u}\mathrm{.}$ Thus the whole loop lies in the closed metric ball of radius $L_u\mathrm{/}2$ about $p$.

If $a>-\mathrm{\infty }$, fix $t_0\in (a,b)$ and repeat step 2.1 with $D=t_0-a$ on the tail $a<t<t_0$. The same compact-box argument and the left-endpoint clause of [F17] extend $\gamma$ past $a$, again contradicting maximality. Hence $a=-\mathrm{\infty }$. Since the maximal geodesic was arbitrary, every maximal domain is $\mathbb{R}$, and [F12] makes $(M,g)$ geodesically complete. This also includes $c=0$: then the box has $V=c=0$ and the geodesic is stationary.

Steps 1.1, 1.2, and 3.1 verify the nonempty, connected, boundaryless, and geodesically complete hypotheses of [F18]. Hopf--Rinow therefore proves that $(M,d_g)$ is complete and supplies a minimizing radial geodesic between every two points.

Suppose for contradiction that $\mathrm{inj}(p)>L_u\mathrm{/}2$. By the supremum convention in [F21], there is $\rho \in {\mathcal{R}}_p$ with $\rho >L_u\mathrm{/}2$. Put $U={\exp }_p(B_{\rho }(0_p))$; the definition of ${\mathcal{R}}_p$ makes ${\exp }_p:B_{\rho }(0_p)\to U$ a diffeomorphism. The local distance formula [F22] gives $U\subseteq B_{d_g}(p,\rho )$. Conversely, if $q\in B_{d_g}(p,\rho )$, the minimizing-vector conclusion of [F18] supplies one $v\in T_{p}M$ with ${\exp }_p(v)=q$ and $\mathrm{\mid }v\mathrm{\mid }=d_g(p,q)<\rho$, so $q\in U$. Hence $U=B_{d_g}(p,\rho )\mathrm{.}$ This is a pointwise existential use of Hopf--Rinow, not a simultaneous choice of vectors.

By step 2.2 and $L_u\mathrm{/}2<\rho$, the image of ${\mathrm{\ell }}_u$ lies in $U$. Since $B_{\rho }(0_p)$ is star-shaped and ${\exp }_p$ is a diffeomorphism there, $H(s,t)={\exp }_p\text{\hspace{0.17em}⁣}((1-t){\exp }_p^{-1}({\mathrm{\ell }}_u(s)))$ is well defined in $U$. It equals ${\mathrm{\ell }}_u$ at $t=0$. At $t=1$ its argument is $0_p$, so its value is $p$. Moreover, if $v={\exp }_p^{-1}(p)$, then [F22] gives $\mathrm{\mid }v\mathrm{\mid }=d_g(p,p)=0$, hence $v=0_p$; because ${\mathrm{\ell }}_u(0)={\mathrm{\ell }}_u(1)=p$, the same calculation keeps $s=0,1$ fixed at $p$ throughout the homotopy. Thus $H$ is a based null-homotopy of ${\mathrm{\ell }}_u$ in $U$, contradicting step 1.4. Therefore $\mathrm{inj}(p_u)\le \frac{L_u}{2}=\pi e^{-u}\mathrm{.}$

Pointwise injectivity radii are positive by [F21], while their global infimum is nonnegative and no larger than any $\mathrm{inj}(p_u)$. Since $e^{-u}\to 0$ as $u\to +\mathrm{\infty }$ by [F5], step 6.1 gives $0\le \mathrm{inj}(M)\le {\inf }_{u\in \mathbb{R}}\pi e^{-u}=0.$ Thus $\mathrm{inj}(M)=0$, although $(M,d_g)$ is complete by step 4.1.

This witness is explicitly nonempty and two-dimensional, so empty-, zero-dimensional-, and one-dimensional-manifold variants are not being asserted. The numerical zero case is the proved global infimum in step 7.1, not a zero pointwise radius. Zero-speed geodesics were retained in step 3.1, both finite maximal endpoints were excluded in steps 2.1 and 3.1, and both endpoints of the loop and its based homotopy were checked in steps 1.4 and 6.1. Assumption [A1] is used only through maximal-geodesic existence [F11], the current geodesic-completeness convention [F12], compact-lift continuation [F17], Hopf--Rinow [F18], and the injectivity-radius and local-normal-distance interfaces [F21] and [F22]. The metric, compact-box, loop-length, noncontractibility, and infimum calculations make no further choices. This is a counterexample, not an iff assertion.