If you have a spatial / positive-signature metric associated with a coordinate basis, and you know the Levi-Civita connection to the metric, and you have an associated conformal metric with determinant 1 that is computed by rescaling the coordinate metric, then what is the geodesic of the conformal metric?
What is the Levi-Civita connection associated with the conformal metric?
If it is ia Levi-Civita connection then it is torsion-free, and if it is non-coordinate then it must have commutation.
But if it has commutation then it must have asymmetry or it must have torsion... Therefore it must have asymmetry.
The conformal metric is calculated by rescaling, and the associated Levi-Civita connection is symmetric.
However the Levi-Civita connection of the conformal metric would not be symmetric, and therefore could not be a rescaling of the Levi-Civita connection of the origianl metric.
Doesn't BSSN formalism call the 'conformal connection' a rescaling of Levi-Civita connection? Wouldn't this be incorrect?

Here I'm assuming the signature of the metric is positive.

$g_{ab} = e_a \cdot e_b = \partial_a \cdot \partial_b$ for a coordinate metric
$g = det[g_{ab}]$
$\bar{g}_{ab} = \phi g_{ab}$ such that $det[\bar{g}_{ab}] = 1$
Therefore $det[\phi g_{ab}] = \phi^n det[g_{ab}] = \phi^n g = 1$,
for $\phi = g^{-\frac{1}{n}}$ for dimension $n$

The Levi-Civita connection for a basis is:
${\Gamma^a}_{bc} = \frac{1}{2} g^{ad} (e_c (g_{db}) + e_b(g_{dc}) - e_d(g_{bc}) + {c^d}_{bc} + {c^d}_{cb} - {c_{cb}}^d)$
For a coordinate basis this is:
${\Gamma^a}_{bc} = \frac{1}{2} g^{ad} (\partial_c (g_{db}) + \partial_b(g_{dc}) - \partial_d(g_{bc}))$

There is freedom in choosing $\bar{e}_a$ such that $\bar{g}_{ab} = \bar{e}_a \cdot \bar{e}_b$,
however so long as $g \ne 1$ we will have $\bar{e}_a \ne \partial_a$, and we have a non-coordinate basis.
Therefore we have non-zero commutation for the Levi-Civita connection associated with $\bar{e}_a$ and $\bar{g}_{ab}$.

Let $\bar{e}_a = {\bar{e}_a}^I \partial_I$ be the linear combination of coordinate basis that the basis of the conformal metric is represented as.
Then $\bar{g}_{ab} = \bar{e}_a \cdot \bar{e}_b = {\bar{e}_a}^I {\bar{e}_b}^J \partial_I \cdot \partial_J = {\bar{e}_a}^I {\bar{e}_b}^J g_{IJ}$
Hmm, notice I'm using these as sum terms, when I should be representing all this as operators.
Then $1 = \bar{g} = det[\bar{g}_{ab}] = det[ {\bar{e}_a}^I g_{IJ} {\bar{e}_b}^J ] = det[ {\bar{e}_a}^I ]^2 \cdot det[g_{IJ}]$
Let $\bar{e} = det[ {\bar{e}_a}^I ]$
Then $1 = \bar{g} = \bar{e}^2 \cdot g$
Then $\bar{e} = g^{-\frac{1}{2}}$
So ${\bar{e}_a}^I$ can be any linear transformation such that $det[{\bar{e}_a}^I] = det[g_{ab}]^{-\frac{1}{2}}$

Now our commutation fits the general form for commutations of linear combinations of coordinate basis:
${c_{ab}}^c = 2 \bar{e}_{[a} ({\bar{e}_{b]}}^I) {\bar{e}^c}_I$
...for ${\bar{e}^a}_I {\bar{e}_b}^I = \delta^a_b$ and ${\bar{e}^a}_I {\bar{e}_b}^J = \delta_I^J$, i.e. $[{\bar{e}^a}_I]$ is the linear inverse of $[{\bar{e}_a}^I]$.

This has the geodesic (taken from my 'metric-cancelling and torsion-free connection' worksheet):
${\bar{\Gamma}^a}_{bc} = {\Gamma^a}_{bc} + {\bar{e}^a}_I \partial_b ({\bar{e}_c}^I)$

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For one conformal rescaling connection in particular: ${\bar{e}_a}^I = \delta_a^I g^{-\frac{1}{2n}}$.
Then ${\bar{e}^a}_I = \delta^a_I g^{\frac{1}{2n}}$.
Then $det[{\bar{e}_a}^I] = g^{-\frac{1}{2}}$, which fulfills our requirement for the conformal rescaling to have a determinant of 1.
$\partial_b {\bar{e}_c}^I = \partial_b g^{-\frac{1}{2n}} \delta_c^I$
$= -\frac{1}{2n} g^{-1-\frac{1}{2n}} \cdot g g^{uv} \partial_b g_{uv} \delta_c^I$
$= -\frac{1}{2n} g^{-\frac{1}{2n}} g^{uv} \partial_b g_{uv} \delta_c^I$

${\bar{e}^a}_I \partial_b ({\bar{e}_c}^I) = \delta^a_I g^{\frac{1}{2n}} \cdot -\frac{1}{2n} g^{-\frac{1}{2n}} g^{uv} \partial_b g_{uv} \delta_c^I$
$ = -\frac{1}{2n} \delta^a_c g^{uv} \partial_b g_{uv}$

The Levi-Civita connection becomes:
${\bar{\Gamma}^a}_{bc} = {\Gamma^a}_{bc} + {\bar{e}^a}_I \partial_b ({\bar{e}_c}^I)$
$= {\Gamma^a}_{bc} - \frac{1}{2n} \delta^a_c g^{uv} \partial_b g_{uv}$

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By definition of the Levi-Civita connection we know that the connection is symmetric in its 2nd and 3rd indexes.
Here's a question: If we have a symmetric connection, does that ensure that there exists a coordinate basis?
I'll venture to say yes. If we have a Levi-Civita connection associated with a coordinate system then we can pick any one coordinate and reparameterize it by scaling the parameter with the conformal factor and we will have a new coordinate system.