Flat local geometry without completeness

Use SI length coordinates: $\theta ,x^1,x^2,x^3$, the periods $L,\mathrm{\ell }$, and the affine parameters $s$ below are measured in metres; the displayed metric has dimensionless components and its line element has units square metres. Thus the tangent $d\gamma \mathrm{/}ds$ is dimensionless with norm $-1$, while the physical four-velocity is $cd\gamma \mathrm{/}ds$ and the ideal proper-time increment is $ds\mathrm{/}c$, with $c$ in metres per second. For $L,\mathrm{\ell }>0$, consider the flat Lorentzian cylinder $M_L=(\mathbb{R}\mathrm{/}L\mathbb{Z})\times {\mathbb{R}}^3$ and the open slab $S_{\mathrm{\ell }}=(0,\mathrm{\ell })\times {\mathbb{R}}^3$, each with metric $-d{\theta }^2+{\sum }_{i=1}^{3}d(x^i{)}^2$ and future direction ${\mathrm{\partial }}_{\theta }$, where $d\theta$ is induced by the covering coordinate on $\mathbb{R}\mathrm{/}L\mathbb{Z}$. The cylinder curve $\gamma :\mathbb{R}\to M_L$, $\gamma (s)=([s{]}_L,0,0,0)$, is a closed future timelike geodesic. The slab contains a timelike geodesic defined on a finite affine interval that cannot be extended inside the slab. Thus locally Minkowski metrics need not preclude a closed future timelike geodesic or a finite-affine-interval geodesic that cannot be extended within its manifold.