A data-fixing isometric immersion of developments is injective, hence a development embedding; uniqueness of data-fixing embeddings

(Reduction to injectivity at two points.) Let $p,q\in M$ with $\phi (p)=\phi (q)=:z$. We show $p=q$. By [L2] the differential $d{\phi }_p$ is an isomorphism, so $\phi$ restricts to a diffeomorphism of some open neighbourhood $V$ of $p$ onto an open neighbourhood of $z$.

(Two geodesics through $q$ and through $p$.) Choose a future-directed timelike vector $u\in T_{q}M$. By [L1] and [L3] there is a maximal geodesic $\gamma :I_{\gamma }\to M$ of $M$ with $\gamma (0)=q$, ${\gamma }^{\mathrm{\prime }}(0)=u$, which is a timelike curve, is inextendible in $M$, and meets $\mathrm{\Sigma }$ in exactly one parameter value ${\tau }_0$. Its image $\phi \circ \gamma$ is a geodesic of $M^{\mathrm{\prime }}$ with $(\phi \circ \gamma )(0)=z$ and velocity $d{\phi }_q({\gamma }^{\mathrm{\prime }}(0))$; since $d{\phi }_q$ is an isomorphism, define $w:=(d{\phi }_p{)}^{-1}(d{\phi }_q({\gamma }^{\mathrm{\prime }}(0)))\in T_{p}M$, a timelike vector, and let $\sigma :I_{\sigma }\to M$ be the maximal geodesic of $M$ with $\sigma (0)=p$, ${\sigma }^{\mathrm{\prime }}(0)=w$; it is timelike, inextendible in $M$, and meets $\mathrm{\Sigma }$ in exactly one parameter value ${\tau }_1$.

(The two image geodesics are the same maximal geodesic of $M^{\mathrm{\prime }}$.) By [L3] both $\phi \circ \sigma$ and $\phi \circ \gamma$ are geodesics of $M^{\mathrm{\prime }}$; at parameter $0$ both have value $z$ and velocity $d{\phi }_p(w)=d{\phi }_q({\gamma }^{\mathrm{\prime }}(0))$. Let $\zeta :I_{\zeta }\to M^{\mathrm{\prime }}$ be the maximal geodesic of $M^{\mathrm{\prime }}$ with $\zeta (0)=z$, ${\zeta }^{\mathrm{\prime }}(0)=d{\phi }_q({\gamma }^{\mathrm{\prime }}(0))$ ([L3]). By uniqueness of maximal geodesics, $I_{\sigma }\subseteq I_{\zeta }$, $I_{\gamma }\subseteq I_{\zeta }$, $\phi \circ \sigma =\zeta {\mathrm{\mid }}_{I_{\sigma }}$ and $\phi \circ \gamma =\zeta {\mathrm{\mid }}_{I_{\gamma }}$.

(The initial-surface crossing parameters coincide.) By step 1.3 and [L1], $\phi (\sigma ({\tau }_1))=\zeta ({\tau }_1)\in {\mathrm{\Sigma }}^{\mathrm{\prime }}$ and $\phi (\gamma ({\tau }_0))=\zeta ({\tau }_0)\in {\mathrm{\Sigma }}^{\mathrm{\prime }}$, with ${\tau }_1\in I_{\sigma }\subseteq I_{\zeta }$ and ${\tau }_0\in I_{\gamma }\subseteq I_{\zeta }$. The geodesic $\zeta$ has timelike velocity, hence is a causal curve, and by [L3] it is inextendible in $M^{\mathrm{\prime }}$; since ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is a Cauchy set of $M^{\mathrm{\prime }}$ ([L1]), the parameter set ${\zeta }^{-1}[{\mathrm{\Sigma }}^{\mathrm{\prime }}]$ has exactly one element. Hence ${\tau }_0={\tau }_1$ and $\zeta ({\tau }_0)=\zeta ({\tau }_1)$.

(Equal points and velocities on $\mathrm{\Sigma }$.) By step 2.1, $\phi (\sigma ({\tau }_0))=\phi (\gamma ({\tau }_0))$ with both arguments in $\mathrm{\Sigma }$; since $\phi {\mathrm{\mid }}_{\mathrm{\Sigma }}=j^{\mathrm{\prime }}\circ j^{-1}$ is injective ([L1]), $\sigma ({\tau }_0)=\gamma ({\tau }_0)=:x$. Differentiating $\phi \circ \sigma =\zeta =\phi \circ \gamma$ at ${\tau }_0$ gives $d{\phi }_x({\sigma }^{\mathrm{\prime }}({\tau }_0))=d{\phi }_x({\gamma }^{\mathrm{\prime }}({\tau }_0))$, and $d{\phi }_x$ is injective by [L2], so ${\sigma }^{\mathrm{\prime }}({\tau }_0)={\gamma }^{\mathrm{\prime }}({\tau }_0)$. Hence $\sigma$ and $\gamma$ are maximal geodesics with the same value and velocity at ${\tau }_0$; by uniqueness [L3] they agree on the intersection of their intervals, which contains $0$, so $p=\sigma (0)=\gamma (0)=q$.

(Clauses (a) and (b).) Clause (a) is [L2]. For clause (b): step 3.1 shows that $\phi$ is injective; by clause (a) and the inverse function theorem on manifolds, $\phi$ is a local diffeomorphism, hence an open map, so $\phi (M)$ is open in $M^{\mathrm{\prime }}$ and $\phi :M\to \phi (M)$ is a bijective local diffeomorphism, hence a diffeomorphism; it satisfies the remaining clauses of a development embedding by hypothesis.

(Clause (c).) Let ${\psi }_1,{\psi }_2:M\to M^{\mathrm{\prime }}$ be data-fixing isometric immersions and let $M$ be connected. By clause (b), already proved in step 4.1, both maps are development embeddings. The pair $(M,{\psi }_1)$ is a common subdevelopment of $M$ and $M^{\mathrm{\prime }}$, so the embedding-only hypotheses of Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps clause (b) hold for $U=M$ and both maps. At $x=j(\sigma )\in \mathrm{\Sigma }$ they have the same value $j^{\mathrm{\prime }}(\sigma )$ and the same differential; hence by rigidity (clause (a) of that lemma) ${\psi }_1={\psi }_2$ on $M$. Taking ${\psi }_2={\mathrm{i}\mathrm{d}}_M:M\to M$ (which is data-fixing by Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (b)) gives the self-isometry statement.