Admissible domains for outward normal graphs

In the restricted and ambient slice coordinates from [F4], the inclusion of $S$ is $x\mapsto (x,0)$; it is therefore smooth with injective differential and identifies $T_{p}S$ with a two-dimensional subspace of $T_p\mathrm{\Sigma }$. In the product charts from [F5], the base projection is smooth and $T_{(p,0)}(S\times \mathbb{R})=T_{p}S\oplus \mathbb{R}$. In these structures the map $(p,r)\mapsto r{\nu }_p$ is smooth. Pulling back the open time-one domain in [F2] gives an open domain for $T$ containing every $(p,0)$. The constant geodesic gives $T(p,0)=p$. Uniqueness of geodesics, applied to the reparametrization $t\mapsto {\gamma }_{p,{\nu }_p}(rt)$ near $r=0$, gives $T(p,r)={\gamma }_{p,{\nu }_p}(r)$ there, so ${\mathrm{\partial }}_{r}T(p,0)={\nu }_p$. Differentiating the identity on $S$ gives $dT(X,0)=X$. Since $T_p\mathrm{\Sigma }=T_{p}S\oplus \mathbb{R}{\nu }_p$, this differential is invertible. The local conclusion follows from [F3].

Suppose $S$ is compact. The open domain of definition and the open local-diffeomorphism locus contain the compact zero section, so finitely many product neighborhoods give one ${\rho }_0>0$ on which $T$ is defined and locally invertible. If no shrinking were injective, for each integer $j$ there would be distinct $(p_j,r_j),(q_j,s_j)$ with $\mathrm{\mid }{r}_j\mathrm{\mid },\mathrm{\mid }{s}_j\mathrm{\mid }<\min ({\rho }_0,1\mathrm{/}j)$ and equal images. Compactness gives a subsequence with $p_j\to p$ and $q_j\to q$. Continuity and the Hausdorff property imply $p=q$, since the common images converge to both $p$ and $q$. Both pairs then lie in a single local inverse neighborhood of $(p,0)$ for large $j$, contradicting injectivity there. Thus a smaller $\rho$ makes $T$ injective; its local inverses agree and yield a smooth inverse on its open image. The pairs are chosen from the fixed nonempty counterexample sets indexed by $j$, using countable choice in [F6].

The graph map $p\mapsto (p,u(p))$ is a smooth embedding into $\mathrm{\Omega }$ in the sense of [F7]: it is smooth and injective, its inverse on its image is the base projection, and its differential has identity base component. Composition with the diffeomorphism $T$ preserves these properties. By [F7], $F_u(S)$ is an embedded two-submanifold of $\mathrm{\Sigma }$ and $F_u:S\to F_u(S)$ is a diffeomorphism. For every nonzero $V\in T_x{F}_u(S)\subset T_x\mathrm{\Sigma }$, $g_x(V,V)=h_x(V,V)>0$ by [F1]; hence the graph is spacelike by the tangent-plane criterion in [F8]. For compact support $K$, finitely many product neighborhoods of $K\times \{0\}$ give a positive radius $\eta$ contained in $\mathrm{\Omega }$ above $K$. The finite maximum $M={\max }_K\mathrm{\mid }\varphi \mathrm{\mid }$ exists. Taking $\mathrm{\mid }ϵ\mathrm{\mid }M<\eta$ keeps the graph in $\mathrm{\Omega }$; off $K$ it is the zero section. For compact $S$ use $K=S$. The case $M=0$ is immediate. This proves all assertions without a completeness hypothesis on $\mathrm{\Sigma }$.