Let $x^I$ be the Minkowski coordinates
Let $x^u$ be the curved coordinates
I'll also use $x^\bar{a}$ for flat space coordinates and $x^a$ for curved space coordintes
Let $ijklmn$ represent spatial coordinates
Let $0$ or $t$ represent the time coordiante

Let $x^\bar{0} = \phi(x^u)$, $x^\bar{i} = x^i$

The basis is defined as:
${e_u}^\bar{0} = \frac{\partial x^\bar{0}}{\partial x^u} = {x^\bar{0}}_{,u} = \phi_{,u}$
${e_u}^\bar{i} = \frac{\partial x^\bar{i}}{\partial x^u} = {x^\bar{i}}_{,u} = \delta^\bar{i}_u$
${e_u}^{\bar{v}} = \downarrow u(i) \overset{\rightarrow \bar{v}(\bar{j})}{ \left[\matrix{ \phi_{,0} & 0 \\ \phi_{,i} & \delta_i^\bar{j} }\right] }$

The inverse basis is defined as:
${e^t}_\bar{0} = (\phi_{,t})^{-1}$
${e^t}_\bar{i} = -(\phi_{,i}) (\phi_{,t})^{-1}$
${e^u}_\bar{i} = \delta^u_i$

Metric:
$g_{uv} = {e_u}^I {e_v}^J \eta_{IJ}$ $= -{e_u}^\bar{0} {e_v}^\bar{0} + {e_u}^\bar{i} {e_v}^\bar{j} \eta_{\bar{i}\bar{j}}$ $= -\phi_{,u} \phi_{,v} + \delta^i_u \delta^j_v \eta_{ij}$
$g_{tt} = -(\phi_{,t})^2$
$g_{ti} = -\phi_{,t} \phi_{,i}$
$g_{ij} = -\phi_{,i} \phi_{,j} + \delta_{ij}$

Metric inverse:
$g^{tt} = (\phi_{,t})^{-2} (-1 + \Sigma_{k=1}^n (\phi_{,k})^2)$
$g^{it} = -(\phi_{,i}) (\phi_{,t})^{-1}$
$g^{ij} = \delta^{ij}$

Metric partial:
$g_{uv,w} = -(\phi_{,u} \phi_{,v})_{,w}$ $=-\phi_{,uw} \phi_{,v} - \phi_{,u} \phi_{,vw}$

Connection
$\Gamma_{uvw} = \frac{1}{2} (g_{uv,w} + g_{uw,v} - g_{vw,u})$
$= \frac{1}{2} ( -\phi_{,uw} \phi_{,v} - \phi_{,u} \phi_{,vw} -\phi_{,uv} \phi_{,w} - \phi_{,u} \phi_{,wv} +\phi_{,vu} \phi_{,w} + \phi_{,v} \phi_{,wu} )$ $= - \phi_{,u} \phi_{,vw}$

${\Gamma^u}_{vw} = g^{ua} \Gamma_{avw}$ $ = -g^{ua} \phi_{,a} \phi_{,vw}$
${\Gamma^t}_{vw} = -g^{tt} \phi_{,t} \phi_{,vw} - g^{ti} \phi_{,i} \phi_{,vw}$ $ = ( -(\phi_{,t})^{-2} (-1 + \Sigma_{k=1}^n (\phi_{,k})^2) \phi_{,t} + \phi_{,i} (\phi_{,t})^{-1} \phi_{,i} ) \phi_{,vw}$
$ = ( 1 - \Sigma_{k=1}^n (\phi_{,k})^2 + (\phi_{,i})^{2} ) (\phi_{,t})^{-1} \phi_{,vw}$

${\Gamma^i}_{vw} = g^{it} \Gamma_{tvw} + g^{ij} \Gamma_{jvw}$
$ = -g^{it} \phi_{,t} \phi_{,vw} - g^{ij} \phi_{,j} \phi_{,vw}$ $ = (\phi_{,i}) (\phi_{,t})^{-1} \phi_{,t} \phi_{,vw} - \delta^{ij} \phi_{,j} \phi_{,vw}$
$ = \phi_{,i} \phi_{,vw} - \phi_{,i} \phi_{,vw}$
$ = 0$

So this gives the opposite of what I wanted: A controllable connection in the new dimension for vectors at rest in the subspace, and no connection in the new dimension for any vectors - especially those at rest in the new dimension.



Now let's try reversing the definition. Still sticking to holonomic spaces, but defining the basis inverse rather than the basis.
Let $x^t = \phi(x^I)$, $x^i = x^\bar{i}$

basis inverse:
${e^t}_\bar{u} = \frac{\partial x^t}{\partial x^\bar{u}} = \phi_{,\bar{u}}$
${e^i}_\bar{u} = \frac{\partial x^i}{\partial x^\bar{u}} = \delta^i_\bar{u}$
${e^u}_\bar{v} = \downarrow u(i) \overset{ \rightarrow \bar{v}(\bar{j}) }{ \left[\matrix{ (\phi_{,0})^{-1} & 0 \\ -\phi_{,\bar{i}} (\phi_{,0})^{-1} & \delta^i_\bar{j} }\right] }$

basis:
${e_t}^0 = (\phi_{,0})^{-1}$
${e_t}^\bar{i} = 0$
${e_i}^0 = -(\phi_{,\bar{i}}) (\phi_{,0})^{-1}$
${e_i}^\bar{j} = \delta_i^\bar{j}$

Now the metric and metric inverse are switched, just like the basis and basis inverse were switched.
Metric:
$g_{tt} = -(\phi_{,0})^{-2}$
$g_{it} = -\phi_{,\bar{i}} (\phi_{,0})^{-2}$
$g_{ij} = -\phi_{,\bar{i}} \phi_{,\bar{j}} (\phi_{,0})^{-2}$

Metric inverse:
$g^{tt} = -(\phi_{,\bar{0}})^2 + \Sigma_{k=1}^n (\phi_{,\bar{k}})^2$
$g^{ti} = \phi_{,\bar{i}}$
$g^{ij} = \delta^{ij}$

Metric partials wrt flat space coordinates:

$g_{tt,\bar{u}} = 2 \phi_{,0\bar{u}} (\phi_{,0})^{-3}$

$g_{ti,\bar{u}} = - \phi_{,\bar{i}\bar{u}} (\phi_{,0})^{-2} + 2 \phi_{,\bar{i}} \phi_{,\bar{u}} (\phi_{,0})^{-3}$
$= ( 2 \phi_{,\bar{i}} \phi_{,\bar{u}} - \phi_{,\bar{i}\bar{u}} \phi_{,0} ) (\phi_{,0})^{-3}$

$g_{ij,\bar{u}} = - (\phi_{,\bar{i}\bar{u}} \phi_{,\bar{j}} + \phi_{,\bar{i}} \phi_{,\bar{j}\bar{u}}) (\phi_{,0})^{-2} + 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{u}} (\phi_{,0})^{-3}$
$= ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{u}} - \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}\bar{u}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}\bar{u}} ) (\phi_{,0})^{-3}$

Metric partials wrt curved space coordinates:

$g_{tt,t} = g_{tt,\bar{u}} {e_t}^\bar{u} = g_{tt,0} {e_t}^0 + g_{tt,\bar{i}} {e_t}^\bar{i}$
$= 2 \phi_{,00} (\phi_{,0})^{-3} \cdot (\phi_{,0})^{-1}$
$= 2 \phi_{,00} (\phi_{,0})^{-4}$

$g_{tt,i} = g_{tt,\bar{u}} {e_i}^\bar{u}$ $= g_{tt,0} {e_i}^0 + g_{tt,\bar{j}} {e_i}^{\bar{j}}$
$= 2 \phi_{,00} (\phi_{,0})^{-3} \cdot -(\phi_{,\bar{i}}) (\phi_{,0})^{-1} + 2 \phi_{,0\bar{j}} (\phi_{,0})^{-3} \cdot \delta^\bar{j}_i$
$= -2 \phi_{,\bar{i}} \phi_{,00} (\phi_{,0})^{-4} + 2 \phi_{,0\bar{i}} (\phi_{,0})^{-3}$
$= 2 (\phi_{,0\bar{i}} \phi_{,0} - \phi_{,\bar{i}} \phi_{,00}) (\phi_{,0})^{-4} $

$g_{ti,t} = g_{ti,\bar{u}} {e_t}^\bar{u} = g_{ti,0} {e_t}^0 + g_{ti,\bar{j}} {e_t}^\bar{j}$
$= ( 2 \phi_{,\bar{i}} \phi_{,0} - \phi_{,0\bar{i}} \phi_{,0} ) (\phi_{,0})^{-3} \cdot (\phi_{,0})^{-1} + ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} - \phi_{,\bar{i}\bar{j}} \phi_{,0} ) (\phi_{,0})^{-3} \cdot 0$
$= ( 2 \phi_{,\bar{i}} - \phi_{,0\bar{i}} ) (\phi_{,0})^{-3}$

$g_{ti,j} = g_{ti,\bar{u}} {e_j}^\bar{u} = g_{ti,0} {e_j}^0 + g_{ti,\bar{k}} {e_j}^\bar{k}$
$= ( 2 \phi_{,\bar{i}} \phi_{,0} - \phi_{,0\bar{i}} \phi_{,0} ) (\phi_{,0})^{-3} \cdot -\phi_{,\bar{j}} (\phi_{,0})^{-1} + (2 \phi_{,\bar{i}} \phi_{,\bar{k}} - \phi_{,\bar{i}\bar{k}} \phi_{,0} ) (\phi_{,0})^{-3} \cdot \delta_j^{\bar{k}} $
$= ( \phi_{,\bar{j}} \phi_{,0\bar{i}} \phi_{,0} - 2 \phi_{,\bar{j}} \phi_{,\bar{i}} \phi_{,0} ) (\phi_{,0})^{-4} + ( 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} - \phi_{,\bar{i}\bar{j}} (\phi_{,0})^2 ) (\phi_{,0})^{-4} $
$= ( \phi_{,\bar{j}} \phi_{,0\bar{i}} - \phi_{,0} \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-3} $

$g_{ij,t} = g_{ij,\bar{u}} {e_t}^\bar{u} = g_{ij,0} {e_t}^0 + g_{ij,\bar{k}} {e_t}^\bar{k}$
$= ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}0} - \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}0} ) (\phi_{,0})^{-3} \cdot (\phi_{,0})^{-1} + ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} - \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}} ) (\phi_{,0})^{-3} \cdot 0$
$= ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}} ) (\phi_{,0})^{-4}$

$g_{ij,k} = g_{ij,\bar{u}} {e_k}^\bar{u} = g_{ij,0} {e_k}^0 + g_{ij,\bar{l}} {e_k}^\bar{l}$
$= ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}} ) (\phi_{,0})^{-3} \cdot -\phi_{,\bar{k}} (\phi_{,0})^{-1} + ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{l}} - \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}\bar{l}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}\bar{l}} ) (\phi_{,0})^{-3} \cdot \delta^\bar{l}_k$
$= \phi_{,\bar{k}} ( - 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} + \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}} + \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}} ) (\phi_{,0})^{-4} + ( 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} - (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}} - (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}} ) (\phi_{,0})^{-4} $
$= ( - 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}} + \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,0\bar{i}} + 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} - (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}} - (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}} ) (\phi_{,0})^{-4} $

Christoffel symbols of the 1st kind:

$\Gamma_{ttt} = \frac{1}{2} g_{tt,t} $ $= \phi_{,00} (\phi_{,0})^{-4}$

$\Gamma_{tti} = \frac{1}{2} (g_{tt,i} + g_{ti,t} - g_{ti,t}) = \frac{1}{2} g_{tt,i}$
$= (\phi_{,0\bar{i}} \phi_{,0} - \phi_{,\bar{i}} \phi_{,00}) (\phi_{,0})^{-4} $

$\Gamma_{tij} = \frac{1}{2} (g_{ti,j} + g_{tj,i} - g_{ij,t})$
$= \frac{1}{2} ( ( \phi_{,\bar{j}} \phi_{,0\bar{i}} - \phi_{,0} \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-3} + ( \phi_{,\bar{i}} \phi_{,0\bar{j}} - \phi_{,0} \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-3} - ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}} ) (\phi_{,0})^{-4} )$
$= ( - \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} + \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}} + \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}} - (\phi_{,0})^2 \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-4}$

$\Gamma_{itt} = \frac{1}{2} (g_{it,t} + g_{it,t} - g_{tt,i}) = g_{it,t} - \frac{1}{2} g_{tt,i}$
$= ( 2 \phi_{,\bar{i}} - \phi_{,0\bar{i}} ) (\phi_{,0})^{-3} - ( \phi_{,0\bar{i}} \phi_{,0} - \phi_{,\bar{i}} \phi_{,00} ) (\phi_{,0})^{-4} $
$= ( 2 \phi_{,0} \phi_{,\bar{i}} - 2 \phi_{,0} \phi_{,0\bar{i}} + \phi_{,\bar{i}} \phi_{,00} ) (\phi_{,0})^{-4} $

$\Gamma_{ijt} = \frac{1}{2} (g_{ij,t} + g_{it,j} - g_{jt,i})$
$= \frac{1}{2} ( ( 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}} - \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}} ) (\phi_{,0})^{-4} + ( \phi_{,\bar{j}} \phi_{,0\bar{i}} - \phi_{,0} \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-3} - ( \phi_{,\bar{i}} \phi_{,0\bar{j}} - \phi_{,0} \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-3} )$
$= \phi_{,\bar{i}} ( \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,0\bar{j}} ) (\phi_{,0})^{-4}$

$\Gamma_{ijjk} = \frac{1}{2} (g_{ij,k} + g_{ik,j} - g_{jk,i})$
$= \frac{1}{2} ( ( - 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}} + \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,0\bar{i}} + 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} - (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}} - (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}} ) (\phi_{,0})^{-4} + ( - 2 \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,\bar{j}} \phi_{,00} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} + \phi_{,0} \phi_{,\bar{k}} \phi_{,\bar{j}} \phi_{,0\bar{i}} + 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}} - (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{k}\bar{j}} - (\phi_{,0})^2 \phi_{,\bar{k}} \phi_{,\bar{i}\bar{j}} ) (\phi_{,0})^{-4} - ( - 2 \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,\bar{i}} \phi_{,00} + \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}} \phi_{,0\bar{k}} + \phi_{,0} \phi_{,\bar{k}} \phi_{,\bar{i}} \phi_{,0\bar{j}} + 2 \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,0\bar{i}} - (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{k}\bar{i}} - (\phi_{,0})^2 \phi_{,\bar{k}} \phi_{,\bar{j}\bar{i}} ) (\phi_{,0})^{-4} )$
$= ( - (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} - \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00} ) (\phi_{,0})^{-4}$

Christoffel symbols of the 2nd kind:
${\Gamma^i}_{tt} = g^{iu} \Gamma_{utt} = g^{it} \Gamma_{ttt} + g^{ij} \Gamma_{jtt}$
$= \phi_{,\bar{i}} \cdot \phi_{,00} (\phi_{,0})^{-4} + \delta^{ij} \cdot ( 2 \phi_{,0} \phi_{,\bar{j}} - 2 \phi_{,0} \phi_{,0\bar{j}} + \phi_{,\bar{j}} \phi_{,00} ) (\phi_{,0})^{-4}$
$= ( 2 \phi_{,0} \phi_{,\bar{i}} - 2 \phi_{,0} \phi_{,0\bar{i}} + 2 \phi_{,\bar{i}} \phi_{,00} ) (\phi_{,0})^{-4}$

${\Gamma^i}_{tj} = g^{iu} \Gamma_{utj} = g^{it} \Gamma_{ttj} + g^{ik} \Gamma_{ktj}$
$= \phi_{,\bar{i}} (\phi_{,0\bar{j}} \phi_{,0} - \phi_{,\bar{j}} \phi_{,00}) (\phi_{,0})^{-4} + \delta^{ik} \phi_{,\bar{k}} ( \phi_{,\bar{j}} \phi_{,00} - \phi_{,0} \phi_{,0\bar{j}} ) (\phi_{,0})^{-4} $
$= 0$

${\Gamma^i}_{jk} = g^{iu} \Gamma_{ujk} = g^{it} \Gamma_{tjk} + g^{il} \Gamma_{ljk}$
$ = \phi_{,\bar{i}} ( - \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00} + \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{k}} + \phi_{,0} \phi_{,\bar{k}} \phi_{,0\bar{j}} - (\phi_{,0})^2 \phi_{,\bar{j}\bar{k}} ) (\phi_{,0})^{-4} + \delta^{il} ( - (\phi_{,0})^2 \phi_{,\bar{l}} \phi_{,\bar{j}\bar{k}} + \phi_{,0} \phi_{,\bar{l}} \phi_{,\bar{k}} \phi_{,0\bar{j}} + \phi_{,0} \phi_{,\bar{l}} \phi_{,\bar{j}} \phi_{,0\bar{k}} - \phi_{,\bar{l}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00} ) (\phi_{,0})^{-4} $
$ = 2 ( - (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}} + \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}} - \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00} ) (\phi_{,0})^{-4} $

TODO check the other geodesic components

Now let $\phi_{,0} = 1$
${\Gamma^i}_{tt} = 2 \phi_{,\bar{i}}$

So let $\phi(t,x^\bar{i}) = \bar{t} - \frac{1}{2} \psi(x^\bar{i})$
Then $\phi_{,0} = 1$
And $\phi_{,\bar{i}} = -\frac{1}{2} \psi_{,\bar{i}}$
Therefore ${\Gamma^i}_{tt} = -\psi_{,\bar{i}}$