The null screen quotient is a smooth positive two-plane

Fix $p\in H$. Nondegeneracy of $g_p$ and $k_p\mathrm{\ne }0$ imply that the linear functional $v\mapsto g_p(k_p,v)$ is nonzero, so choose $v\in T_p\mathcal{M}$ with $g_p(k_p,v)=-1$. In a neighborhood in $H$, set $n:=N\mathrm{/}\sqrt{-g(N,N)}$ and $\alpha :=-g(k,n)>0$, so $n$ and $\alpha$ are smooth; define $v_0:=n\mathrm{/}\alpha$ and $\mathrm{\ell }:=v_0+\frac{1}{2}g(v_0,v_0)k$. Then $g(k,\mathrm{\ell })=g(k,v_0)=-1$, and $g(\mathrm{\ell },\mathrm{\ell })=g(v_0,v_0)+g(v_0,v_0)g(v_0,k)+\frac{1}{4}g(v_0,v_0{)}^{2}g(k,k)=0,$ since $g(v_0,k)=-1$ and $g(k,k)=0$. This construction is smooth locally and gives a null field $\mathrm{\ell }$ paired with $k$ by $-1$.

Let $P_p:={\mathrm{span}}_{\mathbb{R}}\{k_p,{\mathrm{\ell }}_p\}$. Its Gram matrix in the ordered basis $(k_p,{\mathrm{\ell }}_p)$ is $\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right)$, with determinant $-1$ and eigenvalues $1,-1$, so it is nondegenerate with inertia $(1,1,0)$. Therefore $T_p\mathcal{M}=P_p\oplus P_p^{\perp }$; in a basis adapted to this orthogonal direct sum, the Gram matrix is block diagonal. The full inertia is $(3,1,0)$ by [F1], so [F3] makes the restriction to the two-dimensional $P_p^{\perp }$ have inertia $(2,0,0)$, hence be positive definite.

Every $X\in k_p^{\perp }$ has a unique decomposition $X=a{k}_p+b{\mathrm{\ell }}_p+z$ with $z\in P_p^{\perp }$. Pairing with $k_p$ gives $0=g(k_p,X)=-b$, so $b=0$. Hence $k_p^{\perp }=\mathbb{R}{k}_p\oplus P_p^{\perp }$, and the quotient map restricts to a linear isomorphism from $P_p^{\perp }$ to ${\mathcal{S}}_p$. Changing a representative by a multiple of $k_p$ does not change $g_p(X,Y)$ because every representative lies in $k_p^{\perp }$; under the isomorphism the quotient form is the positive-definite restriction to $P_p^{\perp }$ from step 2.1. This proves its dimension, well-definedness and positivity.

The local fields $k,\mathrm{\ell }$ are linearly independent and their $2\times 2$ Gram matrix is invertible. Choose two smooth local vector fields $w_1,w_2$ whose values at $p$ complete them to a basis. Solving the two linear equations $g(k,e_i)=g(\mathrm{\ell },e_i)=0$ for the $k,\mathrm{\ell }$ components of $w_i$ uses the inverse of that smooth invertible Gram matrix, so yields smooth fields $e_1,e_2\in P^{\perp }$; after shrinking the neighborhood they remain linearly independent and form a frame of $P^{\perp }$. By step 3.1 their quotient classes frame $\mathcal{S}$, so the quotient is a smooth rank-two vector bundle and $\bar{g}$ is a smooth positive-definite metric. Applying the two finite Gram–Schmidt operations to $e_1,e_2$ gives a local smooth $\bar{g}$-orthonormal screen frame: each denominator is the positive square root of a smooth positive squared norm, hence smooth by [F4].

The screen form is independent of representatives. Replacing $X$ by $X+fk$ changes $g({\mathrm{\nabla }}_{X}k,Y)$ by $fg({\mathrm{\nabla }}_{k}k,Y)=0$ because $k$ is affine. Replacing $Y$ by $Y+hk$ changes it by $hg({\mathrm{\nabla }}_{X}k,k)=\frac{h}{2}X(g(k,k))=0$, since $\mathrm{\nabla }$ is metric-compatible and $k$ is null on $H$. Thus $B$ descends to a smooth bilinear form on $\mathcal{S}$; smoothness follows from its smooth local-frame components.