• 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; $g_X = \iota^* g_Y$.