• Definitions / Notation used
• The single technical move: affine symmetry
• Metric as induced
• Why this matters
• Key takeaway
• Technical takeaway

If geometry is about how vectors move, then transport comes first. Metrics only summarize the outcome. GU takes this seriously: it treats connections as the primary objects and metrics as derived data.

§Definitions / Notation used

• \(X = X^4\) (observed spacetime), \(Y = Y^{14}\) (ambient/observerse space); immersion \(\iota: X \to Y\); pullback \(\iota^\*\).
• Tangent/normal split along \(\iota(X)\): \(TY|\_X \simeq TX \oplus N_\iota,\) with \(\mathrm{rank}(N_\iota)=10;\) indices \(\mu,\nu\) for \(TX;\) \(a,b\) for \(N_\iota;\) \(M,N\) for \(Y.\)
• Metric: \(g_X := \iota^\* g_Y;\) metric split ansatz \(g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b;\) Hodge stars \(\*_X\) and \(\*_Y.\)
• Gauge structure: \(H\) gauge group; \(\mathcal{N} := \Omega^1(Y, \mathrm{ad}(P_H));\) inhomogeneous group \(\mathcal{G} := H \ltimes \mathcal{N};\) \(\omega = (\varepsilon, \phi);\) background connection \(A_0;\) rotated connection \(B_\omega := A_0 \cdot \varepsilon;\) curvature \(F_B\).

§The single technical move: affine symmetry

The space of connections is affine. Differences of connections live in \(\mathcal{N} = \Omega^1(Y, \mathrm{ad})\).

Gauge transformations alone (\(H\)) do not act transitively on this space. Translations in \(\mathcal{N}\) are missing.

Therefore the symmetry group must be enlarged to

$$\mathcal{G} = H \ltimes \mathcal{N}$$

The background connection \(A_0\) selects a tilted copy of \(H\) inside \(\mathcal{G}\) that fixes \(A_0\). Quotienting by this subgroup identifies the space of connections with \(\mathcal{N}\), coordinatized by \(T\).

This is not an aesthetic choice; it is forced by the geometry of connection space.

§Metric as induced

Once connections are primary, the metric on \(X\) cannot be fundamental. It arises from the immersion \(\iota\) and the ambient metric \(g_Y\):

$$g\_X := \iota^\* g\_Y$$

All gravitational dynamics on \(X\) must therefore descend from transport data on \(Y\).

§Why this matters

This shift explains why gravity and gauge fields can be treated uniformly. Both are aspects of a single connection on \(Y\), acted on by a single symmetry group. The price is abandoning metric primacy—but that is exactly what removes the two-geometries obstruction.

§Key takeaway

Connections form an affine space. Respecting that structure forces \(\mathcal{G} = H \ltimes \mathcal{N}\) and makes the metric secondary.

§Technical takeaway

Connection space \(\simeq \mathcal{G} / \tilde{H}(A_0)\); torsion \(T\) is the \(\mathcal{N}\)-coordinate; \(g\_X = \iota^\* g\_Y\).