Compact metric families have a common spacelike collar with fixed boost rate

Positive definiteness is open and uniform on the compact set $[0,1]\times S$, so a common neighborhood $U$ in $\widehat{\mathrm{\Sigma }}$ is spacelike; restrict to a relatively compact one-sided neighborhood in $U\cap {\mathrm{\Sigma }}_{+}$. To construct the future normal, project $N_q$ orthogonally onto $T\widehat{\mathrm{\Sigma }}$ using the inverse positive induced matrix, subtract this projection, and normalize the remaining timelike vector. This remainder has negative pairing with $N_q$, so normalization selects the stipulated future normal smoothly in $q$ and position. At $S$, local boundary defining functions increasing into ${\mathrm{\Sigma }}_{+}$ give ${\bar{\nu }}_q$ by normalizing their induced gradients; on overlaps these give the same inward unit vector. These operations are smooth, and compactness bounds every fixed finite order in fixed finite charts.

At $p\in S$, the coordinate formula $({\mathrm{\nabla }}_v^{(q)}K{)}^a=v^b{\mathrm{\partial }}_b{K}^a+{\mathrm{\Gamma }}^a{}_{bc}{v}^b{K}^c$ reduces to $v^b{\mathrm{\partial }}_b{K}^a$. Thus $B_p$ is connection independent. To derive skewness, differentiate the coordinate pullback formula $({\mathrm{\Phi }}_t^{\ast }{g}_q{)}_{ab}(x)=g_{q,cd}({\mathrm{\Phi }}_t(x)){\mathrm{\partial }}_a{\mathrm{\Phi }}_t^c(x){\mathrm{\partial }}_b{\mathrm{\Phi }}_t^d(x)$ at $t=0$. The flow has ${\mathrm{\partial }}_t{\mathrm{\Phi }}_t{\mathrm{\mid }}_{t=0}=K$ and ${\mathrm{\partial }}_t{\mathrm{\partial }}_a{\mathrm{\Phi }}_t^c{\mathrm{\mid }}_{t=0}={\mathrm{\partial }}_a{K}^c$, giving $0=({\mathcal{L}}_K{g}_q{)}_{ab}=K^c{\mathrm{\partial }}_c{g}_{q,ab}+g_{q,cb}{\mathrm{\partial }}_a{K}^c+g_{q,ac}{\mathrm{\partial }}_b{K}^c$. Since $K_p=0$, this says exactly $g_q(B_{p}u,v)+g_q(u,B_{p}v)=0$ at $p$. As $B_p$ kills $T_{p}S$, it consequently preserves $(T_{p}S{)}^{{\perp }_{g_q}}$. This derives the needed skewness without using a lemma whose hypothesis already assumes a positive boost matrix.

The induced connection has Christoffel coefficients ${\bar{\mathrm{\Gamma }}}^k{}_{ij}=\frac{1}{2}{\bar{h}}_q^{k\mathrm{\ell }}({\mathrm{\partial }}_i{\bar{h}}_{q,j\mathrm{\ell }}+{\mathrm{\partial }}_j{\bar{h}}_{q,i\mathrm{\ell }}-{\mathrm{\partial }}_{\mathrm{\ell }}{\bar{h}}_{q,ij})$, which are smooth jointly in $q$ and position. Direct substitution verifies metric compatibility and torsion freedom. Apply the coordinate geodesic equation and smooth ODE dependence to initial data $(p,{\bar{\nu }}_q(p))$ to obtain jointly smooth $E_q(p,r)$ on a common short interval, first in $\widehat{\mathrm{\Sigma }}$ for signed $r$. The map $(q,p,r)\mapsto (q,E_q(p,r))$ has invertible differential at $(q,p,0)$: its spatial differential is $(X,a)\mapsto X+a{\bar{\nu }}_q(p)$. The inverse theorems therefore give smooth local inverses including $q$. Compactness gives a common small radius with nonsingular differential. It also gives injectivity for each $q$ at one common radius: otherwise distinct colliding data with radii tending to zero have a subsequence with $q\to q_0$ and base points converging; continuity forces both base limits to be the same point of $S$, contradicting the one local inverse there. For small positive radii the inward initial derivative puts the curves in ${\mathrm{\Sigma }}_{+}$, and compactness makes this uniform. A small $q=0$ normal tube now provides a common one-sided collar $V$ with compact closure in $U\cap {\mathrm{\Sigma }}_{+}$. Choose $2ϵ$ within the common radius, small enough that all the $E_q$ images for $0\le r\le 2ϵ$ lie in $V$. The parameterized inverse neighborhoods of the compact set $[0,1]\times S$ contain $[0,1]\times W$ for a common smaller collar $W$, chosen as a still smaller $q=0$ normal tube with compact closure in $V$; shrink it so all its inverse radii are less than $ϵ$. Both $q=0$ tubes are smooth collars in the sense of [F5]: the restricted maps $c(p,r)=E_0(p,r)$ are embeddings with $c(p,0)=p$, and the signed local inverses identify their half-tube images with open neighborhoods of $S$ in ${\mathrm{\Sigma }}_{+}$. Smoothness and compactness also give the asserted fixed-order bounds for the exponential maps and their inverses.

At $p\in S$, the pair $({\bar{n}}_q,{\bar{\nu }}_q)$ is an orthonormal basis of $(T_{p}S{)}^{{\perp }_{g_q}}$, with metric matrix $\mathrm{diag}(-1,1)$. Skewness from step 1.2 forces $B_p{\bar{\nu }}_q=a(q,p){\bar{n}}_q$ and $B_p{\bar{n}}_q=a(q,p){\bar{\nu }}_q$. This normal plane identifies with $T_{p}M\mathrm{/}{T}_{p}S$, so the eigenvalue hypothesis gives $a(q,p{)}^2={\kappa }^2$. The normals depend smoothly on $q$, hence $a(q,p)$ is continuous. For each fixed $p$ it cannot change sign on $[0,1]$; the stated inward-basis positivity gives $a(1,p)=\kappa$, so $a(q,p)=\kappa$ throughout. In particular ${\mathrm{\nabla }}_{{\bar{\nu }}_q}^{(q)}K=\kappa {\bar{n}}_q$ on $S$, with the inward, not the outward boundary normal of ${\mathrm{\Sigma }}_{+}$.

Write $D$ for the induced connection just constructed. Along a normal geodesic let $\dot{\gamma }={\mathrm{\partial }}_r{E}_q$ and let $J$ be a variation in its base point. Metric compatibility and the geodesic equation give ${\bar{h}}_q(\dot{\gamma },\dot{\gamma })=1$. Torsion freedom and the commuting variation parameters give $D_{\dot{\gamma }}J=D_J\dot{\gamma }$, so $\frac{d}{dr}{\bar{h}}_q(\dot{\gamma },J)={\bar{h}}_q(\dot{\gamma },D_J\dot{\gamma })=\frac{1}{2}J{\bar{h}}_q(\dot{\gamma },\dot{\gamma })=0$. Its initial value is zero because ${\bar{\nu }}_q\perp TS$. Thus the pulled-back metric is $d{r}^2+h_{q,r}$. Every curve from $S$ staying in this normal tube has length at least its terminal radius. A curve in $V$ that exits the radius-$2ϵ$ tube must first reach that radius, costing at least $2ϵ$; compactness of $S$ and the embedding on the closed smaller tubes exclude other exit points. For points of $W$, whose radius is less than $ϵ$, such an exit cannot improve on the radial geodesic of length $r$. This proves $r=d_{{\bar{h}}_q,V}(\cdot ,S)$ on $W$, and ${\mathrm{\nabla }}^{{\bar{h}}_q}{\rho }_q={\mathrm{\partial }}_r{E}_q$ is the desired smooth extension of the inward boundary normal, with the stated uniform bounds.

In fixed ambient charts along the geodesics of step 2.1, Taylor expansion of $K$, with $K{\mathrm{\mid }}_S=0$ and initial derivative from step 2.2, gives $K=\kappa r{\bar{n}}_q{\mathrm{\mid }}_{r=0}+O(r^2)$ in coordinate components. Smoothness replaces the base normal by the normal at the endpoint at a cost $O(r^2)$, so $K=\kappa {\rho }_q{\bar{n}}_q+O({\rho }_q^2)$ as a field at that point. Smoothness on compact parameter and coordinate domains makes the remainder uniform. Using $g_q({\bar{n}}_q,{\bar{n}}_q)=-1$ and uniformly bounded metric components gives $g_q(K,K)=-{\kappa }^2{\rho }_q^2+O({\rho }_q^3)$ and $g_q(K,{\bar{n}}_q)=-\kappa {\rho }_q+O({\rho }_q^2)$. Since $\kappa >0$, a common shrinking makes the first expression negative and the second negative for ${\rho }_q>0$. Thus $K$ is timelike and future directed on the selected punctured collar, with all claimed uniformity.