The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric

(Smoothness and Lorentzian signature.) In a product chart $(x^0,x^1,x^2,x^3)$ with $x^0=t$ the components of $g$ are the smooth functions displayed in FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter: $g_{00}=-c^2$, $g_{0i}=0$, $g_{ij}=a(t{)}^2{\gamma }_{ij}(x)$. Product charts cover $M$ and their transition maps are products of the transition maps of $I$ and $\mathrm{\Sigma }$, so by the transformation law [L1] the family $g$ is a smooth symmetric $(0,2)$-tensor field on $M$. At a point, the matrix of $g_p$ in the coordinate basis is block diagonal, $\mathrm{diag}(-c^2,a(t{)}^2({\gamma }_{ij}(x){)}_{i,j=1}^3)$. The $1\times 1$ block has the single negative entry $-c^2$, and the $3\times 3$ block is a positive multiple of the positive-definite matrix of ${\gamma }_x$; hence the diagonal entries of a diagonalization are one negative and three positive numbers, and $g_p$ is nondegenerate with inertia $(3,1,0)$. By [L2] this makes $g_p$ a Lorentz form at every point; by Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model the smooth field $g$ with these pointwise Lorentz forms is a Lorentzian metric of signature $(-,+,+,+)$ on $M$. Since $g(N,N)=g_{00}=-c^2<0$ at every point, [L3] makes $N={\mathrm{\partial }}_t$ a time orientation and $(M,g,N)$ a time-oriented smooth Lorentzian four-manifold.

(The comoving congruence and the comoving curves.) By step 1.1 the manifold $(M,g,N)$ is time-oriented smooth Lorentzian, so the causal vocabulary of [L3] applies. The field $n=c^{-1}{\mathrm{\partial }}_t$ is a smooth vector field; its metric pairing is $g(n,n)=c^{-2}g({\mathrm{\partial }}_t,{\mathrm{\partial }}_t)=c^{-2}(-c^2)=-1$, and $g(n,N)=-c<0$. Hence $n$ is a future-directed unit timelike vector field, i.e. a smooth future-directed unit timelike congruence. For fixed $x\in \mathrm{\Sigma }$ the smooth curve ${\gamma }_x:I\to M$, ${\gamma }_x(t)=(t,x)$ has tangent ${\mathrm{\partial }}_t=cn$; its image is $I\times \{x\}$ and its velocity satisfies $g({\gamma }_x^{\mathrm{\prime }},{\gamma }_x^{\mathrm{\prime }})=-c^2<0$ and $g({\gamma }_x^{\mathrm{\prime }},N)=-c^2<0$, so it is a future timelike worldline. For $(t_0,x)\in M$, the curve ${\eta }_{t_0,x}(s)=(t_0+s\mathrm{/}c,x)$ on $c(I-t_0)$ has derivative $c^{-1}{\mathrm{\partial }}_t=n$, so it is the integral curve of $n$ through that point by [L5]. Any extension would still satisfy $dt\mathrm{/}ds=c^{-1}$ and constant spatial coordinates, hence would leave $I\times \mathrm{\Sigma }$ beyond this interval; its domain is therefore maximal. The curve ${\gamma }_x$ is the cosmic-time reparametrization $s=c(t-t_0)$ of this same observer trajectory, with velocity $cn$.

(Cosmic time is proper time.) For the comoving curve ${\gamma }_x$ of step 2.1, the proper-time density of [L4] is $h(t)=\sqrt{-g({\gamma }_x^{\mathrm{\prime }}(t),{\gamma }_x^{\mathrm{\prime }}(t))}=\sqrt{c^2}=c$, a constant. Hence for $t_0<t_1$ in $I$, the proper-time increment is $c^{-1}{\int }_{t_0}^{t_1}cdt=t_1-t_0$ by [L4]; in particular the parameter $t$ is proper time along ${\gamma }_x$ up to the choice of zero. The physical four-velocity, defined as the derivative of the worldline with respect to this proper time (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent), is therefore $U=d{\gamma }_x\mathrm{/}dt={\mathrm{\partial }}_t=cn$, with $g(U,U)=-c^2$.

(Slices, orthogonality and the scaled spatial curvature.) Fix $t_0\in I$. The map ${\iota }_{t_0}:\mathrm{\Sigma }\to M$, ${\iota }_{t_0}(x)=(t_0,x)$, is a smooth embedding onto the embedded submanifold $\{t_0\}\times \mathrm{\Sigma }$. In a product chart the differential of ${\iota }_{t_0}$ sends ${\mathrm{\partial }}_i{\mathrm{\mid }}_x$ to ${\mathrm{\partial }}_i{\mathrm{\mid }}_{(t_0,x)}$, so the pullback components are $({\iota }_{t_0}^{\ast }g{)}_{ij}=g_{ij}(t_0,x)=a(t_0{)}^2{\gamma }_{ij}(x)$: a smooth, symmetric, positive definite field, hence a Riemannian metric ${\gamma }_{t_0}=a(t_0{)}^2\gamma$ by [L6]. Moreover $g(n,{\mathrm{\partial }}_i)=0$ for $i=1,2,3$ because $g_{0i}=0$ and $n$ has no spatial component: the congruence is orthogonal to every slice. Finally, by [L7] the constant rescaling gives $\mathrm{Ric}({\gamma }_{t_0})=\mathrm{Ric}(\gamma )=2k\gamma =\frac{2k}{a(t_0{)}^2}{\gamma }_{t_0}$, so ${\gamma }_{t_0}$ satisfies the Ricci form with parameter $k\mathrm{/}a(t_0{)}^2$; the equivalence [L7] turns this into constant sectional curvature $k\mathrm{/}a(t_0{)}^2$.

(Conclusion.) Steps 1.1 and 2.1 establish clauses (a) and (b), step 3.1 establishes the proper-time and four-velocity statements of (b), and step 3.2 establishes clause (c). The proofs used only the displayed component functions, the pointwise algebra of symmetric bilinear forms and the cited general facts; no field equation and no choice beyond the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of the suppliers was used.