Normal boost action and invariant two-forms at a fixed surface

For $p\in S$, the constant curve at $p$ is an integral curve of $K$ because $K_p=0$. Uniqueness makes ${\mathrm{\Phi }}_t(p)=p$ throughout the interval on which the flow at $p$ is defined. Since the flow fixes every point of $S$, differentiating its restriction to $S$ gives $d{\mathrm{\Phi }}_t{\mathrm{\mid }}_{p}X=X$ for $X\in T_{p}S$.

For the circle average, the flow law makes ${\varphi }_s$ a diffeomorphism with inverse ${\varphi }_{-s}$. Because it preserves $S$ and $g$, each transported vector in the integrand is normal to $S$ at $p$; it is future timelike by the stated cone preservation. In a future orthonormal frame at $p$, each such vector has time component $a(s)>\mathrm{\mid }b(s)\mathrm{\mid }$, where $b(s)$ denotes its spatial components. Continuity on $[0,2\pi ]$ gives a positive minimum of $a(s)-\mathrm{\mid }b(s)\mathrm{\mid }$; the triangle inequality for integrals therefore gives $\bar{a}>\mathrm{\mid }\bar{b}\mathrm{\mid }$. Thus the average is future timelike and, by linearity, normal.

Put $V(s,p)=d{\varphi }_s{\mathrm{\mid }}_{{\varphi }_{-s}(p)}{n}_{{\varphi }_{-s}(p)}$. Smoothness of the action and $n$ makes the components $V^a(s,x)$ smooth jointly in $s$ and the slice coordinates $x$ on $S$, with the output expressed in one ambient chart around $p$; the moving source point can use different charts. Choose a closed coordinate rectangle $Q$ contained in this slice chart, with $p$ in its interior. On $[0,2\pi ]\times Q$ all derivatives of $V^a$ are continuous. Fixing the other coordinates, apply F6 on a compact rectangle in $(x_i,s)$, and repeat for every partial derivative. For each multi-index $\alpha$ this gives ${\mathrm{\partial }}_x^{\alpha }{\bar{n}}^a(x)=(2\pi {)}^{-1}{\int }_0^{2\pi }{\mathrm{\partial }}_x^{\alpha }{V}^a(s,x)ds$ in the interior of $Q$. These derivatives are continuous: their difference at $x,y$ is bounded in absolute value by ${\sup }_{s\in [0,2\pi ]}\mathrm{\mid }{\mathrm{\partial }}_x^{\alpha }{V}^a(s,x)-{\mathrm{\partial }}_x^{\alpha }{V}^a(s,y)\mathrm{\mid }$, which tends to zero as $y\to x$ by uniform continuity on the compact rectangle. Hence $\bar{n}$ is smooth near each $p$.

Fix $p\in S$ and put $A(t)=d{\mathrm{\Phi }}_t{\mathrm{\mid }}_p$. Differentiating the flow law at the fixed point gives $A(t+h)=A(h)A(t)$ whenever the times are admissible. In a coordinate chart, differentiating the flow equation at $t=0$ in the initial point gives $A^{\mathrm{\prime }}(0)=D{K}_p$. The connection term in ${\mathrm{\nabla }}_{v}K$ vanishes at $p$ because $K_p=0$, so $A^{\mathrm{\prime }}(0)=B_p$. It follows from the flow law that $A^{\mathrm{\prime }}(t)=B_{p}A(t)$ and $A(0)=I$.

The positive definite induced metric gives the direct sum $T_{p}M=T_{p}S\oplus (T_{p}S{)}^{\perp }$. Differentiating $A(t)X=X$ from step 1.1 at zero and using step 2.1 gives $B_{p}X=0$ for $X\in T_{p}S$. On the normal plane, the stated matrix of $B_p$ has square ${\kappa }_p^{2}I$. Let $C(t)$ be the displayed hyperbolic matrix on that plane and extend it to $\tilde{C}(t)$ on $T_{p}M$ by the identity on $T_{p}S$. Since $B_p$ preserves the normal plane and vanishes on $T_{p}S$, direct block-matrix differentiation gives $\tilde{C}(0)=I$, ${\tilde{C}}^{\mathrm{\prime }}(t)=B_p\tilde{C}(t)=\tilde{C}(t)B_p$, and $\tilde{C}(-t)\tilde{C}(t)=I$. Thus the full-space product satisfies $\frac{d}{dt}(\tilde{C}(-t)A(t))=-\tilde{C}(-t)B_{p}A(t)+\tilde{C}(-t)B_{p}A(t)=0$. Its value at zero is $I$, so $A(t)=\tilde{C}(t)$ on the interval of definition. In particular, $A(t)$ preserves the normal plane, where it equals $C(t)$, and is the identity on $T_{p}S$.

Set $v_{+}=e_0+e_1$ and $v_{-}=e_0-e_1$. Step 3.1 gives $A(t)v_{\pm }=e^{\pm {\kappa }_{p}t}{v}_{\pm }$. For $X\in T_{p}S$, invariance of $F$, the pullback rule and step 1.1 give $F_p(v_{\pm },X)=({\mathrm{\Phi }}_t^{\ast }F{)}_p(v_{\pm },X)=F_p(A(t)v_{\pm },A(t)X)=e^{\pm {\kappa }_{p}t}{F}_p(v_{\pm },X)$. The flow domain contains an interval about zero, so a nonzero admissible $t$ exists; since ${\kappa }_p>0$, each exponential factor differs from one. Thus both values vanish, and $v_{+},v_{-}$ span the normal plane.

The flow law gives $d{\varphi }_t{\mathrm{\mid }}_{p}V(s,p)=V(s+t,{\varphi }_t(p))$. The integrand $V$ is $2\pi$-periodic in $s$, so substitution $u=s+t$ in its full-period integral gives $d{\varphi }_t{\mathrm{\mid }}_p{\bar{n}}_p={\bar{n}}_{{\varphi }_t(p)}$. This uses only $2\pi$-periodicity, so shorter common periods and the identity flow are included. The denominator in the normalization is smooth and strictly positive by step 1.2, and is preserved by the isometries. Consequently $\widehat{n}$ is smooth, future unit timelike, normal, and invariant.