A nonzero degenerate generator cannot vanish at a regular bifurcation section

Along one affine generator choose a coordinate component $L^j$ nonzero at $\lambda =0$ and shrink the interval so it remains nonzero. Since $K$ is in the line spanned by $L$, the scalar $f=K^j\mathrm{/}{L}^j$ satisfies $K=fL$. The quotient is smooth up to zero, $f(0)=0$, and $f\mathrm{\ne }0$ on $(0,ϵ)$. Smoothness of such a quotient follows by iterating the quotient rule; the same argument in each coordinate direction applies to the local quotients below. The extension-independent curve derivative and affine equation give ${\mathrm{\nabla }}_{K}K=f{D}_{\lambda }(fL)=f{f}^{\mathrm{\prime }}L$.

For the bifurcate-horizon consequence, choose one branch $H$ and a point $q$ of its spacelike bifurcation surface $S$. In a slice chart write $H=\{r=0\}$ with $dr\mathrm{\ne }0$, and set $L=(dr{)}^{\mathrm{♯}}{\mathrm{\mid }}_H$ using the inverse metric. The annihilator of $TH=\ker dr$ is $\mathbb{R}dr$, so $(TH{)}^{\perp }=\mathbb{R}L$ and $L\mathrm{\ne }0$. Because the tangent metric has a nonzero radical, this orthogonal line lies in $TH$; thus $L$ is tangent and null. Choose a component $L^j$ nonzero near $q$ and put $f=K^j\mathrm{/}{L}^j$ there. On $H_{\ast }$ normality gives $K=fL$; at a zero of $K$ both sides vanish, so this equality holds on the entire patch. The quotient argument of step 1.1 makes $f$ smooth, including at its zeros.

In the affine case, degeneracy and step 1.1 imply $f{f}^{\mathrm{\prime }}=0$. Since $f\mathrm{\ne }0$ on $(0,ϵ)$, $f^{\mathrm{\prime }}=0$ there; F5 applies to that interval and makes $f$ constant. Its constant value is nonzero, whereas continuity at zero forces it to equal $f(0)=0$. This proves the first assertion.

On the branch patch of step 2.1, let $X$ be any smooth tangent field. Metric compatibility, torsion freeness, and the tangency of $[L,X]$ give $g({\mathrm{\nabla }}_{L}L,X)=-g(L,{\mathrm{\nabla }}_{L}X)=-g(L,{\mathrm{\nabla }}_{X}L+[L,X])=-\frac{1}{2}X(g(L,L))=0$. Here $g(L,X)$ and $g(L,L)$ vanish on $H$, so their tangential derivatives vanish; slice-chart extensions justify the ambient calculation and F2 makes its tangential values independent of the extensions. Thus ${\mathrm{\nabla }}_{L}L$ lies in $(TH{)}^{\perp }=\mathbb{R}L$, and ${\mathrm{\nabla }}_{L}L=aL$ for a smooth scalar $a$, obtained by division by the same nonzero component $L^j$. This establishes the pregeodesic property locally rather than presuming an affine ruling.

Apply F6 to $L$ on the three-manifold $H$ to obtain smooth integral curves $\mathrm{\Phi }(t,p)$ for $p$ in a small patch of $S$, with $\mathrm{\Phi }(0,p)=p$. The nonzero null vector $L_q$ cannot lie in the positive-definite plane $T_{q}S$, so $T_{q}H=T_{q}S\oplus \mathbb{R}{L}_q$. The differential of $(t,p)\mapsto \mathrm{\Phi }(t,p)$ at $(0,q)$ is therefore the isomorphism $(b,v)\mapsto b{L}_q+v$. The inverse function theorem, followed by restriction to a product $(-\delta ,\delta )\times W$ with $q\in W\subset S$, makes this map a diffeomorphism onto a nonempty open collar $V$ in $H$. Shrink the product so its image stays in the patch above.

On that patch, the connection rules and step 3.1 yield ${\mathrm{\nabla }}_{K}K=f(L(f)+af)L$. On $H_{\ast }$, $f\mathrm{\ne }0$ and degeneracy implies $L(f)+af=0$. This scalar is smooth on the whole patch; density of $H_{\ast }$ in $H$ forces it to vanish everywhere there, since a nonzero value would by continuity have a nonempty open neighborhood avoiding $H_{\ast }$. This uses density on $H$, without asserting density or nonvanishing on each individual generator.

For each $p\in W$, set $h_p(t)=f(\mathrm{\Phi }(t,p))$ and $a_p(t)=a(\mathrm{\Phi }(t,p))$. The chain rule and step 4.1 give $h_p^{\mathrm{\prime }}=-a_p(t)h_p$ with $h_p(0)=0$, because $K=0$ on $S$. The coefficient $a_p$ is smooth, so the right-hand side is continuous and locally Lipschitz in the scalar state on $(-\delta ,\delta )\times \mathbb{R}$. The zero function is another solution; F7 gives $h_p=0$ on the common interval $(-\delta ,\delta )$. Hence $K=0$ throughout $V$, contradicting density of $H_{\ast }$. No regular bifurcation surface exists for this same degenerate generator. The qualification concerning another specified Killing field remains intact.