Projective reduction recovers metric transport

4. The unrestricted affine connection

4.1 Fix the curvature convention before varying

For unrestricted $\Gamma^\alpha{}_{\mu\beta}$, equation (1) gives

$$R_{\mu\nu}(A)= \partial_\alpha\Gamma^\alpha{}_{\nu\mu}-\partial_\nu\Gamma^\alpha{}_{\alpha\mu} +\Gamma^\alpha{}_{\alpha\rho}\Gamma^\rho{}_{\nu\mu} -\Gamma^\alpha{}_{\nu\rho}\Gamma^\rho{}_{\alpha\mu}. \tag{17}$$

Order matters. Extending the familiar symmetric-affinity contraction $\Gamma^\alpha{}_{\mu\nu}\Gamma^\beta{}_{\alpha\beta}-\Gamma^\alpha{}_{\beta\mu}\Gamma^\beta{}_{\alpha\nu}$ without its lower-index symmetry gives a different polynomial. For $\Gamma^\alpha{}_{\mu\beta}=\delta^\alpha_\beta\omega_\mu$ that polynomial is $(n-1)\omega_\mu\omega_\nu$, whereas the quadratic term of (17) vanishes.

Define the symmetric operators

$$D_{\mu\nu}(c)=\operatorname{Sym}_{\mu\nu} (\partial_\alpha c^\alpha{}_{\nu\mu}-\partial_\nu c^\alpha{}_{\alpha\mu}), \tag{18}$$ $$Q_{\mu\nu}(c)=\operatorname{Sym}_{\mu\nu} (c^\alpha{}_{\alpha\rho}c^\rho{}_{\nu\mu}-c^\alpha{}_{\nu\rho}c^\rho{}_{\alpha\mu}). \tag{19}$$

Then $R_{(\mu\nu)}(\lambda c)=\lambda D_{\mu\nu}(c)+\lambda^2Q_{\mu\nu}(c)$. With symmetric lower indices these reduce to Deser's contractions.

4.2 The full affine Hessian has just the projective kernel

Take $\int B^{\mu\nu}R_{\mu\nu}(A)$ with nondegenerate metric density and matter independent of $A$. Write

$$A=\mathrm{LC}(g)+C,\qquad C\in\Gamma(TU\otimes T^*U\otimes T^*U). \tag{20}$$

Since $\nabla^g B=0$, the linear terms in $\nabla^g C$ integrate to the matched boundary current. The bulk distortion action is

$$q_B(C)=B^{\mu\nu} \left(C^\alpha{}_{\alpha\rho}C^\rho{}_{\nu\mu} -C^\alpha{}_{\nu\rho}C^\rho{}_{\alpha\mu}\right). \tag{21}$$

All non-gauge distortions are auxiliary variables. There is no new connection wave.

Theorem 4.1 (General affine stationary connection). For smooth nondegenerate metric density in dimension $n\gt2$, the stationary points of (21) are exactly

$$C^\alpha{}_{\mu\beta}=\delta^\alpha_\beta\omega_\mu. \tag{22}$$

The Hessian has rank $n^3-n$ independently of signature. With metric matter the connection equation gives $A=\mathrm{LC}(g)+I\omega$.

Proof. At a point divide by $v$ and vary the independent component $C^\ell{}_{pq}$. The equation is

$$\delta^p_\ell g^{mn}C^q{}_{nm}+g^{pq}C^a{}_{a\ell} -g^{mp}C^q{}_{\ell m}-g^{qn}C^p{}_{n\ell}=0. \tag{23}$$

Lower the first index of $C$, and put $U_a=g^{bc}C_{abc}$ and $V_b=g^{ac}C_{cab}$. Lowering $p,q$ gives

$$C_{abc}+C_{cab}=g_{bc}U_a+g_{ca}V_b. \tag{24}$$

Combine the three cyclic equations to obtain

$$2C_{abc}=g_{bc}(U_a-V_a)+g_{ca}(U_b+V_b)+g_{ab}(V_c-U_c). \tag{25}$$

Its $b,c$ trace gives $(n-2)(U_a-V_a)=0$. Thus $U=V$ and $C_{abc}=g_{ac}V_b$, which is (22). Conversely direct substitution satisfies each equation. The kernel of the quadratic form has exactly the $n$ projective directions, which proves the rank. No definiteness enters the argument.

The classical projective equivalence leaves raw torsion $\delta^\alpha_\nu\omega_\mu-\delta^\alpha_\mu\omega_\nu$ and permits a non-metric-compatible raw connection. It does not say the unrestricted stationary connection is uniquely Levi–Civita.

4.3 Restore volume in the transport readout

Under a projective shift,

$$A_\mu\mapsto A_\mu+\omega_\mu I,\qquad F_{\mu\nu}\mapsto F_{\mu\nu}+(\partial_\mu\omega_\nu-\partial_\nu\omega_\mu)I . \tag{26}$$

Ricci changes by an antisymmetric tensor, so the symmetric-density action is invariant. The transport changes by

$$T_\gamma\mapsto e^{-\int_\gamma\omega}T_\gamma . \tag{27}$$

Bare open-vector transport therefore does not descend. A concrete observable is needed. Set $v_B=|\det B|^{1/(n-2)}$ and, in the prepared endpoint frames, define

$$T_\gamma^B= \left[\frac{v_B(x)}{v_B(y)\det T_\gamma}\right]^{1/n}T_\gamma , \qquad \gamma:x\to y . \tag{28}$$

Transports generated continuously from identity have positive determinant, so the positive root is unambiguous. In four physical dimensions $v_B=\sqrt{-\det B}$.

Theorem 4.2 (Volume-restored transport). On connected coordinate patches the map $(B,T)\mapsto(B,T^B)$ is projectively invariant, multiplicative and natural under diffeomorphic coordinate changes, including orientation reversal. Its infinitesimal connection is

$$A^B_\mu=A_\mu-\frac{\operatorname{tr}A_\mu}{n}I +\frac{\partial_\mu\log v_B}{n}I . \tag{29}$$

The readout identifies the projective quotient with the regular families satisfying $\det T_\gamma^B=v_B(x)/v_B(y)$. On the connection equations $A^B=\mathrm{LC}(g_B)$, and $T^B$ is exactly metric parallel transport.

Proof. The scalar in (27) cancels its $n$th power in the determinant in (28). Intermediate volumes and determinants cancel for successive paths. Under a pullback with endpoint Jacobians $J_x,J_y$, $T'=J_y^{-1}TJ_x$ and $v'(x)=|\det J_x|v(\Phi x)$. The sign of $\det J$ is constant along the path, giving $\det J_x/\det J_y=|\det J_x|/|\det J_y|$. The multiplier is invariant and $T^B$ transforms as a tangent map.

Differentiate (28) on a short path, using $d\log\det T/ds=-\operatorname{tr}(A_\mu)\dot\gamma^\mu$. This gives (29), with trace $\partial_\mu\log v_B$. Each projective class has this unique representative, obtained by subtracting $(\operatorname{tr}A-d\log v_B)/n$. The inverse transport theorem gives the quotient and volume-family inverse identities. On the connection equation $A=\mathrm{LC}(g_B)+I\omega$ and $\operatorname{tr}\mathrm{LC}(g_B)=d\log v_B$, so (29) removes exactly $\omega$.

Equation (29) fixes trace, not off-shell torsion. The unrestricted quotient has 60 connection coordinates in four dimensions; the torsion-free parent has 40. The extra twenty coordinates disappear through the algebraic equations. The two parents agree at the on-shell metric endpoint but have different off-shell field spaces.

The readouts are $B,T^B$ and metric matter. Retaining bare transport or a projectively sensitive boundary norm prevents this quotient. A trace-fixed boundary collar is an alternative with transformations restricted to those preserving it. These are observable and boundary contracts, not consequences of admission.

The normalization uses endpoint densities and the determinant of the supplied transport. It does not insert a Levi–Civita formula into the source kinematics. Variation may occur before quotienting. If the volume section moves during variation, its trace contributes a projective tangent, whose contraction with the connection Euler derivative vanishes by the trace identity below.