Local data and global descent

Proof. (1) The metric. Condition (15) says exactly that the intrinsic tensors $g_i$ and $g_j$ restrict to the same tensor field on $U_i\cap U_j$. Since $\Gamma(S^2T^*-)$ is a sheaf on $M$, the gluing axiom yields a unique global section $g\in\Gamma(S^2T^*M)$ with $g|_{U_i}=g_i$ for all $i$. Nondegeneracy and signature are pointwise properties of the symmetric bilinear form $g_x$ on $T_xM$; every $x\in M$ lies in some $U_i$, where $g_x=(g_i)_x$ is Lorentzian by hypothesis. Hence $g$ is Lorentzian. (If one prefers to present the data in coordinate components, the equivalent hypothesis is (13), which expresses that the $\check g_i$ are the component arrays of a single intrinsic tensor; the invariance of signature under the linear isomorphism $\mathrm{d}\psi_{ij}(x)$ is then Sylvester’s law of inertia.)

(2) The bundle. Form the disjoint union $\coprod_{i\in I}U_i\times G\times\{i\}$ and impose $$(x,a,i)\;\sim\;\bigl(x,\;h_{ji}(x)\,a,\;j\bigr), \qquad x\in U_i\cap U_j,\; a\in G .\tag{18}$$ Reflexivity follows from $h_{ii}=1$, symmetry from $h_{ij}=h_{ji}^{-1}$, and transitivity from $h_{kj}h_{ji}=h_{ki}$; hence (18) is an equivalence relation. Let $P$ be the quotient with the quotient topology, $\pi[(x,a,i)]:=x$.

We record the trivializations rigorously. Fix $i$ and $x\in U_i$. Every equivalence class lying over $x$ has exactly one representative whose third entry is $i$: existence, because if a class contains $(x,b,j)$ then it contains $(x,h_{ij}(x)b,i)$; uniqueness, because $(x,a,i)\sim(x,a',i)$ forces $a'=h_{ii}(x)a=a$. Hence the map $$\varphi_i\colon \pi^{-1}(U_i)\longrightarrow U_i\times G, \qquad \varphi_i\bigl([(x,a,i)]\bigr)=(x,a)\tag{19}$$ is a well-defined bijection, with inverse $\varphi_i^{-1}(x,a)=[(x,a,i)]$; it is a homeomorphism by definition of the quotient topology, and we give $P$ the smooth structure for which the $\varphi_i$ are diffeomorphisms (consistent, since the change of trivialization $\varphi_i\circ\varphi_j^{-1}(x,a)=(x,h_{ij}(x)a)$ is smooth). The right $G$-action $[(x,a,i)]\cdot b:=[(x,ab,i)]$ is well defined, because $h_{ji}$ acts by left multiplication and hence commutes with right multiplication; it is free and transitive on fibres, and each $\varphi_i$ is $G$-equivariant. Thus $\pi\colon P\to M$ is a principal $G$-bundle with transition functions $h_{ij}$.

(3) The connection. Let $s_i\colon U_i\to P$ be the canonical local sections $s_i(x):=\varphi_i^{-1}(x,1)$, so that $s_j=s_i\cdot h_{ij}$ on $U_i\cap U_j$. A principal connection $A\in\Omega^1(P,\mathfrak{g})$ is determined by its local representatives $A_i:=s_i^*A$, and a family $\{A_i\}$ arises from a (necessarily unique) global $A$ if and only if it obeys the law induced by $s_j=s_i\cdot h_{ij}$, namely $$s_j^*A \;=\; \operatorname{Ad}_{h_{ij}^{-1}}\bigl(s_i^*A\bigr)+h_{ij}^*\theta_G,$$ which is precisely (16). Given (16), define $A$ over $U_i$ by the pullback identity $$\bigl(\varphi_i^{-1}\bigr)^{*}A \;=\; \operatorname{Ad}_{a^{-1}}\bigl(\pi_U^{*}A_i\bigr)\;+\;\theta_G \qquad\text{on}\qquad U_i\times G,\tag{20}$$ where $\pi_U\colon U_i\times G\to U_i$ and the projection $U_i\times G\to G$ are understood, $a$ denotes the $G$-coordinate, and $\theta_G$ is pulled back along that projection. Formula (20) is an equality of $\mathfrak{g}$-valued $1$-forms on the manifold $U_i\times G$; it is not an evaluation at a point. Two such local definitions agree on $\pi^{-1}(U_i\cap U_j)$: transporting (20) for $j$ through $\varphi_i\circ\varphi_j^{-1}(x,a)=(x,h_{ij}(x)a)$ and using the standard identity $(L_{h}\circ)^*\theta_G=\theta_G$ together with the Maurer–Cartan expansion of $\mathrm{d}(h_{ij}a)$ reproduces (20) for $i$ exactly when (16) holds. The resulting $1$-form is $G$-equivariant, $R_b^*A=\operatorname{Ad}_{b^{-1}}A$, and reproduces the fundamental vector fields, $A(\zeta_X)=X$; hence it is a principal connection [6].

(4) The matter section. Sections of $P\times_\rho W$ correspond bijectively to $\rho$-equivariant maps $\tilde\Phi\colon P\to W$ with $\tilde\Phi(p\cdot a)=\rho(a^{-1})\tilde\Phi(p)$, and such maps correspond to families $\Phi_i:=\tilde\Phi\circ s_i$ obeying $\Phi_j=\tilde\Phi\circ(s_i\cdot h_{ij})=\rho(h_{ij}^{-1})\Phi_i$, which is (17). Hence the $\Phi_i$ glue to a unique global section $\Phi$.

Uniqueness up to isomorphism. Suppose two systems of local trivializations over a common cover (or over covers admitting a common refinement, after passing to it) produce compatible data $\{g_i,h_{ij},A_i,\Phi_i\}$ and $\{g_i,h'_{ij},A'_i,\Phi'_i\}$ with cohomologous cocycles, $h'_{ij}=\lambda_i^{-1}h_{ij}\lambda_j$ for smooth $\lambda_i\colon U_i\to G$, together with $A'_i=\operatorname{Ad}_{\lambda_i^{-1}}A_i+\lambda_i^*\theta_G$ and $\Phi'_i=\rho(\lambda_i^{-1})\Phi_i$. Then the maps $\varphi_i'^{-1}(x,a)\mapsto\varphi_i^{-1}(x,\lambda_i(x)a)$ agree on overlaps, exactly because the cocycles are cohomologous, and assemble into a $G$-equivariant bundle isomorphism $P'\to P$ covering $\mathrm{id}_M$, carrying $A'$ to $A$ and $\Phi'$ to $\Phi$. That is, it is a vertical arrow of $\mathfrak{G}_{\mathrm{geo}}$. Conversely, an arrow of $\mathfrak{G}_{\mathrm{geo}}$ between global tuples, read in local trivializations, exhibits the two systems of local data as related in exactly this way. Hence the global tuple is well defined precisely as an object of $\mathfrak{G}_{\mathrm{geo}}$ up to isomorphism.

Converse. Given a global tuple, choose any cover $\{U_i\}$ trivializing $P$ (possible since $M$ is paracompact and $P$ is locally trivial); the induced $g_i:=g|_{U_i}$, $h_{ij}$, $A_i=s_i^*A$, $\Phi_i$ satisfy (14)–(17) by the computations above. ◻

Proof. Immediate from Theorem 3.4: any global configuration restricts to compatible local data, so failure of compatibility for every presentation of the proposed values is an obstruction to the existence of a configuration presenting them. Membership in $\mathcal{C}_{\mathrm{geo}}$ is a further restriction, since Theorem 3.4 produces a global tuple but does not certify that it satisfies the declared causal, boundary, or topological conditions. ◻