A flat Lorentzian four-manifold that is not isometric to Minkowski space: the rolled-up flat model R^4 / Z by a timelike translation

(The action.) In the chart $x$ the action reads $(x^0,x^1,x^2,x^3)\mapsto (x^0+nL,x^1,x^2,x^3)$; each $p\mapsto n\cdot p$ is a smooth translation, so the action is smooth; it is an action of the group $\mathbb{Z}$ on $\widehat{M}$ with $0\cdot p=p$ and $(n+m)\cdot p=n\cdot (m\cdot p)$. It is free: $n\cdot p=p$ forces $nL=0$, hence $n=0$. It preserves $\widehat{g}$: the differential of a translation in the chart is the identity, so ${\widehat{g}}_{n\cdot p}(d(n\cdot {)}_{p}V,d(n\cdot {)}_{p}W)={\widehat{g}}_p(V,W)$ by [L1]. It is proper: if $K_1,K_2\subseteq \widehat{M}$ are compact and $p\in K_1$ with $n\cdot p\in K_2$, then $\mathrm{\mid }n\mathrm{\mid }L\le {\sup }_{K_1}\mathrm{\mid }{x}^0\mathrm{\mid }+{\sup }_{K_2}\mathrm{\mid }{x}^0\mathrm{\mid }<\mathrm{\infty }$ by [L8], so only finitely many $n$ occur; hence ${\mathrm{\Theta }}^{-1}(K_1\times K_2)$ is a finite union of compact sets, one for each such $n$, and is compact. Finally $\widehat{N}={\mathrm{\partial }}_0$ is $\mathrm{\Gamma }$-invariant with $\widehat{g}(\widehat{N},\widehat{N})=-1<0$, since it is invariant under translations.

(The smooth quotient.) By step 1.1, [L2] and the zero-dimensional Lie-group structure of [L3], $M$ is a Hausdorff second-countable smooth four-manifold and $q$ is a smooth surjective submersion. Thus $d{q}_p$ is onto between four-dimensional tangent spaces and hence invertible for every $p$. The inverse function theorem in [L2] gives a local diffeomorphism $q{\mathrm{\mid }}_{\widehat{U}}:\widehat{U}\to V$ at each point, with smooth inverse $s:V\to \widehat{U}$. The quotient is connected as the image of connected $\widehat{M}$ under the continuous surjection $q$.

(Metric and time orientation.) For $r=q(p)$ and $v,w\in T_{r}M$ define $g_r(v,w):={\widehat{g}}_p((d{q}_p{)}^{-1}v,(d{q}_p{)}^{-1}w)$ and $N_r:=d{q}_p({\widehat{N}}_p)$. These definitions are independent of the representative: if $p^{\mathrm{\prime }}=n\cdot p$, differentiating $q\circ (n\cdot )=q$ gives $d{q}_{p^{\mathrm{\prime }}}\circ d(n\cdot {)}_p=d{q}_p$, so the inverse images of $v,w$ at $p^{\mathrm{\prime }}$ are the translates of those at $p$. Metric invariance and field invariance from step 1.1 therefore give the same values for $g_r$ and $N_r$. The form $g_r$ is symmetric and nondegenerate with signature $(-,+,+,+)$ because $d{q}_p$ is invertible and ${\widehat{g}}_p$ has that signature. Over every inverse section $s$ of step 2.1 one has $g{\mathrm{\mid }}_V=s^{\ast }\widehat{g}$ and $N_r=d{q}_{s(r)}({\widehat{N}}_{s(r)})$, both smooth by their chart formulas. Thus $g$ is a smooth Lorentzian metric with $q^{\ast }g=\widehat{g}$; this identity also uniquely determines $g$, since $q$ is onto and $d{q}_p$ is invertible. Furthermore $g(N,N{)}_{q(p)}=\widehat{g}(\widehat{N},\widehat{N}{)}_p=-1$, so $N$ is a time orientation.

(The closed curve.) The curve $\widehat{\sigma }(s):=x^{-1}(s,0,0,0)$ on $[0,L]$ is smooth and satisfies $\widehat{\sigma }(L)=x^{-1}(L,0,0,0)=1\cdot \widehat{\sigma }(0)$. Hence $\sigma =q\circ \widehat{\sigma }$ is smooth and $\sigma (L)=q(1\cdot \widehat{\sigma }(0))=\sigma (0)$. Its tangent is ${\sigma }^{\mathrm{\prime }}(s)=d{q}_{\widehat{\sigma }(s)}({\mathrm{\partial }}_0)\mathrm{\ne }0$ by step 2.1, and step 3.1 gives $g({\sigma }^{\mathrm{\prime }},{\sigma }^{\mathrm{\prime }})=-1$ and $g(N,{\sigma }^{\mathrm{\prime }})=-1$. Thus $\sigma$ is future timelike and nonconstant with coincident endpoints.

(Flatness.) Let $r\in M$ and $p\in q^{-1}(r)$. Since $q$ is a local diffeomorphism there are open neighborhoods $\widehat{U}$ of $p$ and $V$ of $r$ with $q{\mathrm{\mid }}_{\widehat{U}}:\widehat{U}\to V$ a diffeomorphism; let $s:V\to \widehat{U}$ be its inverse and $u:=x\circ s:V\to {\mathbb{R}}^4$ the induced chart (Smooth manifolds and their smooth charts). For $r^{\mathrm{\prime }}\in V$ and $v,w\in T_{r^{\mathrm{\prime }}}M$ one has $g_{r^{\mathrm{\prime }}}(v,w)={\widehat{g}}_{s(r^{\mathrm{\prime }})}(d{s}_{r^{\mathrm{\prime }}}v,d{s}_{r^{\mathrm{\prime }}}w)$ by $q^{\ast }g=\widehat{g}$ from step 3.1. In the induced chart $u=x\circ s$, $ds$ takes each coordinate basis vector to the corresponding $x$-coordinate basis vector; hence $g_{ab}={\widehat{g}}_{ab}\circ s={\eta }_{ab}$, constant by [L1] and [L5]. By [L6] the Christoffel symbols of the metric-selected connection vanish identically in this chart, and then the curvature formula of [L4] gives $R^a{}_{bcd}(r)=0$. As $r\in M$ was arbitrary and vanishing of a tensor at a point is chart-independent, $R\equiv 0$ on $M$. The same charts exhibit for each point a neighborhood isometric to an open subset of Minkowski space, in accordance with clause (d) of Vanishing curvature on a normal neighborhood is a local isometry to Minkowski space.

(No global isometry.) Suppose $F:M\to {\mathbb{R}}^4$ satisfies $F^{\ast }\eta =g$. Let $f:=(F\circ \sigma {)}^0=x^0\circ F\circ \sigma$; $f$ is smooth by [L5], so continuous on $[0,L]$ and differentiable on $(0,L)$, with $f(L)-f(0)=0$ because $\sigma (L)=\sigma (0)$. By [L7] there is $c\in (0,L)$ with $f^{\mathrm{\prime }}(c)=0$. On the other hand, for every $s$, [L8] and step 4.1 give $\eta ((F\circ \sigma {)}^{\mathrm{\prime }}(s),(F\circ \sigma {)}^{\mathrm{\prime }}(s))=g({\sigma }^{\mathrm{\prime }}(s),{\sigma }^{\mathrm{\prime }}(s))=-1$, that is $-(f^{\mathrm{\prime }}(s){)}^2+{\sum }_{i=1}^3((F\circ \sigma {)}^i{)}^{\mathrm{\prime }}(s{)}^2=-1,$ so $(f^{\mathrm{\prime }}(s){)}^2=1+{\sum }_{i=1}^3((F\circ \sigma {)}^i{)}^{\mathrm{\prime }}(s{)}^2\ge 1$ and $f^{\mathrm{\prime }}(s)\mathrm{\ne }0$; in particular $f^{\mathrm{\prime }}(c)\mathrm{\ne }0$, a contradiction. Hence no such $F$ exists. Since a global chart $u$ with $g_{ab}={\eta }_{ab}$ constant would make $u$ an isometry onto its image (with $\eta$ given by those constants), no such chart exists either.

(Conclusion.) Steps 2.1, 3.1, 4.1 and 4.2 produce a connected time-oriented smooth Lorentzian four-manifold $(M,g,N)$ with $R\equiv 0$ and a nonconstant closed future timelike curve; step 5.1 shows it is not isometric to Minkowski space and admits no global chart with constant components ${\eta }_{ab}$. This refutes the statement; the local isometry statement of Vanishing curvature on a normal neighborhood is a local isometry to Minkowski space is not contradicted but is precisely local, as its clause (d) asserts.