Back

If we define our covariant derivative such that it eliminates the metric:
$0={\mathrm{\nabla }}_c(g_{ab}{e}^a\otimes e^b)$
$0=e_c(g_{ab})e^a\otimes e^b+g_{ab}{\mathrm{\nabla }}_c{e}^a\otimes e^b+g_{ab}{e}^a\otimes {\mathrm{\nabla }}_c{e}^b$
$0=e_c(g_{ab})e^a\otimes e^b-g_{ab}{{\mathrm{\Gamma }}^a}_{cu}{e}^u\otimes e^b-g_{ab}{e}^a\otimes {{\mathrm{\Gamma }}^b}_{cu}{e}^u$
$e_c(g_{ab})e^a\otimes e^b=({\mathrm{\Gamma }}_{bca}+{\mathrm{\Gamma }}_{acb})e^a\otimes e^b$
$e_c(g_{ab})={\mathrm{\Gamma }}_{bca}+{\mathrm{\Gamma }}_{acb}$

Due to this constraint, we can say that all metric-eliminating connections are symmetric possess this identity:
${\mathrm{\Gamma }}_{(a|b|c)}=\frac{1}{2}{e}_b(g_{ac})$

From here we can define our covariant derivative in terms of structure constants and torsion:
$e_a(g_{bc})={\mathrm{\Gamma }}_{cab}+{\mathrm{\Gamma }}_{bac}$
$e_b(g_{ac})={\mathrm{\Gamma }}_{abc}+{\mathrm{\Gamma }}_{cba}$
$e_c(g_{ab})={\mathrm{\Gamma }}_{bca}+{\mathrm{\Gamma }}_{acb}$
...subtract...
$e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc})={\mathrm{\Gamma }}_{bca}+{\mathrm{\Gamma }}_{acb}+{\mathrm{\Gamma }}_{abc}+{\mathrm{\Gamma }}_{cba}-{\mathrm{\Gamma }}_{cab}-{\mathrm{\Gamma }}_{bac}$
$e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc})=2{\mathrm{\Gamma }}_{abc}-c_{acb}-T_{bac}-c_{abc}-T_{cab}-c_{bca}-T_{abc}$
${\mathrm{\Gamma }}_{abc}=\frac{1}{2}(e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc})+c_{abc}+c_{acb}-c_{cba}+T_{cab}+T_{bac}-T_{acb})$

Notice that $\mathrm{\nabla }$ can be defined to have any ${{\mathrm{\Gamma }}^a}_{bc}$, however there is only one ${{\hat{\mathrm{\Gamma }}}^a}_{bc}$ such that the torsion is zero, and that is the Levi-Civita connection. Let $\hat{\mathrm{\nabla }}$ be the covariant derivative associated with the zero-torsion connection $\hat{\mathrm{\Gamma }}$.
${\hat{\mathrm{\Gamma }}}_{abc}=\frac{1}{2}(e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc})+c_{abc}+c_{acb}-c_{cba})$
This is also known as the "Christoffel symbol of the 1st kind".

Misner Thorne Wheeler "Gravitation" list this as ${\hat{\mathrm{\Gamma }}}_{abc}=\frac{1}{2}(e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc})+c_{abc}+c_{acb}-c_{bca})$, so notice the last commutation coefficient is negative'd. This coincides with its reversed definition of ${\mathrm{\nabla }}_b{e}_c={{\mathrm{\Gamma }}^a}_{cb}{e}_a$, which would make sense. If you reverse b and c then the one term that is negative'd is the last commutation term. However, in contrast, Wikipedia here holds my un-reversed definition of ${{\mathrm{\Gamma }}^a}_{bc}$ but still lists the same anholonomic definition. Maybe the authors of the different sections of the page used different conventions?

Christoffel of the 2nd kind / affine connection: ${{\hat{\mathrm{\Gamma }}}^a}_{bc}=g^{ad}{\hat{\mathrm{\Gamma }}}_{dbc}$

in a holonomic basis (so ${c_{ab}}^c=0$):
${\hat{\mathrm{\Gamma }}}_{abc}=\frac{1}{2}(e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc}))$

Levi-Civita connection in a non-coordinate basis:
If $e_a={e_a}^{\tilde{a}}{\partial }_{\tilde{a}}$ and $e^a={e^a}_{\tilde{a}}d{x}^{\tilde{a}}$ are linear combinations of a coordinate basis ${\partial }_{\tilde{a}}$ and dual $d{x}^{\tilde{a}}$ then we can calculate the following:
Using the identities that ${\delta }_b^a={e^a}_{\tilde{u}}{e_b}^{\tilde{u}}$ and ${\delta }_{\tilde{b}}^{\tilde{a}}={e_u}^{\tilde{a}}{e^u}_{\tilde{b}}$
Let ${\tilde{g}}_{\tilde{a}\tilde{b}}={\partial }_{\tilde{a}}\cdot {\partial }_{\tilde{b}}$
Let $g_{ab}=e_a\cdot e_b={e_a}^{\tilde{a}}{\partial }_{\tilde{a}}\cdot {e_b}^{\tilde{b}}{\partial }_{\tilde{b}}={e_a}^{\tilde{a}}\cdot {e_b}^{\tilde{b}}{\tilde{g}}_{\tilde{a}\tilde{b}}$.
Let ${c_{ab}}^c{e}_c=[e_a,e_b]$, so (from the 'structure constants' worksheet on structure constants of a linear transform of a coordinate basis) ${c_{ab}}^c=2e_{[a}({e_{b]}}^{\tilde{c}}){e^c}_{\tilde{c}}$
${\hat{\mathrm{\Gamma }}}_{abc}=\frac{1}{2}(e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc})+c_{acb}+c_{abc}-c_{cba})$
$=\frac{1}{2}(e_c({e_a}^{\tilde{a}}{e_b}^{\tilde{b}}{\tilde{g}}_{\tilde{a}\tilde{b}})+e_b({e_a}^{\tilde{a}}{e_c}^{\tilde{c}}{\tilde{g}}_{\tilde{a}\tilde{c}})-e_a({e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{\tilde{g}}_{\tilde{b}\tilde{c}})+2e_{[a}({e_{c]}}^{\tilde{d}}){e^d}_{\tilde{d}}{g}_{db}+2e_{[a}({e_{b]}}^{\tilde{d}}){e^d}_{\tilde{d}}{g}_{dc}-2e_{[c}({e_{b]}}^{\tilde{d}}){e^d}_{\tilde{d}}{g}_{da})$
$=\frac{1}{2}(e_c({e_a}^{\tilde{a}}){e_b}^{\tilde{b}}{\tilde{g}}_{\tilde{a}\tilde{b}}+{e_a}^{\tilde{a}}{e}_c({e_b}^{\tilde{b}}){\tilde{g}}_{\tilde{a}\tilde{b}}+{e_a}^{\tilde{a}}{e_b}^{\tilde{b}}{e}_c({\tilde{g}}_{\tilde{a}\tilde{b}})+e_b({e_a}^{\tilde{a}}){e_c}^{\tilde{c}}{\tilde{g}}_{\tilde{a}\tilde{c}}+{e_a}^{\tilde{a}}{e}_b({e_c}^{\tilde{c}}){\tilde{g}}_{\tilde{a}\tilde{c}}+{e_a}^{\tilde{a}}{e_c}^{\tilde{c}}{e}_b({\tilde{g}}_{\tilde{a}\tilde{c}})-e_a({e_b}^{\tilde{b}}){e_c}^{\tilde{c}}{\tilde{g}}_{\tilde{b}\tilde{c}}-{e_b}^{\tilde{b}}{e}_a({e_c}^{\tilde{c}}){\tilde{g}}_{\tilde{b}\tilde{c}}-{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{e}_a({\tilde{g}}_{\tilde{b}\tilde{c}})+e_a({e_c}^{\tilde{d}})e_{b\tilde{d}}-e_c({e_a}^{\tilde{d}})e_{b\tilde{d}}+e_a({e_b}^{\tilde{d}})e_{c\tilde{d}}-e_b({e_a}^{\tilde{d}})e_{c\tilde{d}}-e_c({e_b}^{\tilde{d}})e_{a\tilde{d}}+e_b({e_c}^{\tilde{d}})e_{a\tilde{d}})$
$=\frac{1}{2}(e_c({e_a}^{\tilde{a}})e_{b\tilde{a}}+e_c({e_b}^{\tilde{b}})e_{a\tilde{b}}+{e_a}^{\tilde{a}}{e_b}^{\tilde{b}}{e}_c({\tilde{g}}_{\tilde{a}\tilde{b}})+e_b({e_a}^{\tilde{a}})e_{c\tilde{a}}+e_b({e_c}^{\tilde{c}})e_{a\tilde{c}}+{e_a}^{\tilde{a}}{e_c}^{\tilde{c}}{e}_b({\tilde{g}}_{\tilde{a}\tilde{c}})-e_a({e_b}^{\tilde{b}})e_{c\tilde{b}}-e_a({e_c}^{\tilde{c}})e_{b\tilde{c}}-{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{e}_a({\tilde{g}}_{\tilde{b}\tilde{c}})+e_a({e_c}^{\tilde{d}})e_{b\tilde{d}}-e_c({e_a}^{\tilde{d}})e_{b\tilde{d}}+e_a({e_b}^{\tilde{d}})e_{c\tilde{d}}-e_b({e_a}^{\tilde{d}})e_{c\tilde{d}}-e_c({e_b}^{\tilde{d}})e_{a\tilde{d}}+e_b({e_c}^{\tilde{d}})e_{a\tilde{d}})$
$=\frac{1}{2}({e_a}^{\tilde{a}}{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{\partial }_{\tilde{c}}({\tilde{g}}_{\tilde{a}\tilde{b}})+{e_a}^{\tilde{a}}{e_c}^{\tilde{c}}{e_b}^{\tilde{b}}{\partial }_{\tilde{b}}({\tilde{g}}_{\tilde{a}\tilde{c}})-{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{e_a}^{\tilde{a}}{\partial }_{\tilde{a}}({\tilde{g}}_{\tilde{b}\tilde{c}}))+e_b({e_c}^{\tilde{c}})e_{a\tilde{c}}$
${\hat{\mathrm{\Gamma }}}_{abc}={e_a}^{\tilde{a}}{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{\tilde{\mathrm{\Gamma }}}_{\tilde{a}\tilde{b}\tilde{c}}+e_{a\tilde{a}}{e}_b({e_c}^{\tilde{a}})$
${{\hat{\mathrm{\Gamma }}}^a}_{bc}={e^a}_{\tilde{a}}{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{\tilde{b}\tilde{c}}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})$
...where ${{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{\tilde{b}\tilde{c}}$ is the Levi-Civita connection associated with the coordinate basis ${\partial }_{\tilde{a}}$.

From there, of course,
${{\hat{\mathrm{\Gamma }}}^a}_{[bc]}={e^a}_{\tilde{a}}{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{[\tilde{b}\tilde{c}]}+{e^a}_{\tilde{a}}{e}_{[b}({e_{c]}}^{\tilde{a}})$
and a coordinate basis has no commutation, so ${{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{[\tilde{b}\tilde{c}]}=0$
so ${{\hat{\mathrm{\Gamma }}}^a}_{[bc]}={e^a}_{\tilde{a}}{e}_{[b}({e_{c]}}^{\tilde{a}})=\frac{1}{2}{c_{bc}}^a$

If we use un-tilde-d indexes to denote transformation by the ${e_a}^{\tilde{a}}$ basis from the coordinate (tilde) to non-coordinate (non-tilde) basis, then we can represent the transformed coordinate basis Levi-Civita connections as then we can represent this as ${{\tilde{\mathrm{\Gamma }}}^a}_{bc}={e^a}_{\tilde{a}}{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{\tilde{b}\tilde{c}}$,
and the identity becomes:
${{\hat{\mathrm{\Gamma }}}^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^a}_{bc}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})$
Notice how this matches up with the covariant derivative definition: ${{\mathrm{\Gamma }}^a}_{bc}=e^a({\mathrm{\nabla }}_b{e}_c)$

If we want to transforms all this to the coordinate basis:
${e_a}^{\tilde{a}}{e^b}_{\tilde{b}}{e^c}_{\tilde{c}}{{\hat{\mathrm{\Gamma }}}^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{\tilde{b}\tilde{c}}+{e^c}_{\tilde{c}}{e^b}_{\tilde{b}}{e}_b({e_c}^{\tilde{a}})$
${e_a}^{\tilde{a}}{e^b}_{\tilde{b}}{e^c}_{\tilde{c}}{{\hat{\mathrm{\Gamma }}}^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{\tilde{b}\tilde{c}}+{e^c}_{\tilde{c}}{\partial }_{\tilde{b}}({e_c}^{\tilde{a}})$

---

What does the non-coordinate metric-cancelling torsion-free covariant derivative of a object look like when represented with coordinate-basis indexes?
Let $\tilde{\mathrm{\nabla }}$ be the coordinate-metric-cancelling torsion-free covariant-derivative (for coordinate metric ${\tilde{g}}_{\tilde{a}\tilde{b}}$).
Let $\hat{\mathrm{\nabla }}$ be the non-coordinate-metric-cancelling torsion-free covariant-derivative (for non-coordinate metric $g_{ij}$).
From the worksheet on "covariant derivative":
${\tilde{\mathrm{\nabla }}}_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}}{\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}})=({\partial }_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}})+\underset{a_i\leftrightarrow c}{{T^{\tilde{A}}}_{\tilde{B}}}{{\tilde{\mathrm{\Gamma }}}^{a_i}}_{uc}-\underset{b_j\leftrightarrow c}{{T^A}_{\tilde{B}}}{{\tilde{\mathrm{\Gamma }}}^c}_{u{b}_j}){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
So in a non-coordinate basis for a non-coordinate-metric-cancelling torsion-free covariant-derivative:
${\hat{\mathrm{\nabla }}}_u({T^A}_B{e}_A\otimes e^B)=(e_u({T^A}_B)+\underset{a_i\leftrightarrow c}{{T^A}_B}{{\hat{\mathrm{\Gamma }}}^{a_i}}_{uc}-\underset{b_j\leftrightarrow c}{{T^A}_B}{{\hat{\mathrm{\Gamma }}}^c}_{u{b}_j})e_A\otimes e^B$
... using ${{\hat{\mathrm{\Gamma }}}^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^a}_{bc}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})$ (where tilde indexes are coordinate-basis and non-tilde are non-coordinate-basis)
${\hat{\mathrm{\nabla }}}_u({T^A}_B{e}_A\otimes e^B)=(e_u({T^A}_B)+\underset{a_i\leftrightarrow c}{{T^A}_B}({{\tilde{\mathrm{\Gamma }}}^{a_i}}_{uc}+{e^{a_i}}_{{\tilde{a}}_i}{e}_u({e_c}^{{\tilde{a}}_i}))-\underset{b_j\leftrightarrow c}{{T^A}_B}({{\tilde{\mathrm{\Gamma }}}^c}_{u{b}_j}+{e^c}_{\tilde{c}}{e}_u({e_{b_j}}^{\tilde{c}})))e_A\otimes e^B$
...transform all indexes outside of derivatives from non-coordinate-basis to coordinate-basis:
$={e_u}^{\tilde{u}}({\partial }_{\tilde{u}}({T^A}_B)\cdot {e_{a_i}}^{{\tilde{a}}_i}\cdot ...\cdot {e^{b_i}}_{{\tilde{b}}_i}\cdot ...+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}+{\partial }_{\tilde{u}}({e_c}^{{\tilde{a}}_i}){e^c}_{\tilde{c}})-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}+{\partial }_{\tilde{u}}({e_c}^{\tilde{c}}){e^c}_{{\tilde{b}}_j})){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
...transform indexes within partial derivative:
$={e_u}^{\tilde{u}}({\partial }_{\tilde{u}}({T^{\tilde{C}}}_{\tilde{D}}\cdot {e^{a_i}}_{{\tilde{c}}_i}\cdot ...\cdot {e_{b_i}}^{{\tilde{c}}_i}\cdot ...)\cdot {e_{a_i}}^{{\tilde{a}}_i}\cdot ...\cdot {e^{b_i}}_{{\tilde{b}}_i}\cdot ...+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}+{\partial }_{\tilde{u}}({e_c}^{{\tilde{a}}_i}){e^c}_{\tilde{c}})-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}+{\partial }_{\tilde{u}}({e_c}^{\tilde{c}}){e^c}_{{\tilde{b}}_j})){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
...distribute partial through basis transforms of indexes of T:
$={e_u}^{\tilde{u}}(({\partial }_{\tilde{u}}({T^{\tilde{C}}}_{\tilde{D}})\cdot {e^{a_i}}_{{\tilde{c}}_i}\cdot ...\cdot {e_{b_i}}^{{\tilde{c}}_i}\cdot ...+{T^{\tilde{C}}}_{\tilde{D}}\cdot ...\cdot {\partial }_{\tilde{u}}{e^{a_i}}_{{\tilde{c}}_i}\cdot ...\cdot {e_{b_i}}^{{\tilde{c}}_i}\cdot ...+{T^{\tilde{C}}}_{\tilde{D}}\cdot {e^{a_i}}_{{\tilde{c}}_i}\cdot ...\cdot {\partial }_{\tilde{u}}{e_{b_i}}^{{\tilde{c}}_i}\cdot ...)\cdot {e_{a_i}}^{{\tilde{a}}_i}\cdot ...\cdot {e^{b_i}}_{{\tilde{b}}_i}\cdot ...+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}+{\partial }_{\tilde{u}}({e_c}^{{\tilde{a}}_i}){e^c}_{\tilde{c}})-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}+{\partial }_{\tilde{u}}({e_c}^{\tilde{c}}){e^c}_{{\tilde{b}}_j})){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
$={e_u}^{\tilde{u}}({\partial }_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}})+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e^{a_i}}_{\tilde{c}}){e_{a_i}}^{{\tilde{a}}_i}+\underset{{\tilde{b}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_{b_i}}^{\tilde{c}}){e^{b_i}}_{{\tilde{b}}_i}+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}+{\partial }_{\tilde{u}}({e_c}^{{\tilde{a}}_i}){e^c}_{\tilde{c}})-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}({{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}+{\partial }_{\tilde{u}}({e_c}^{\tilde{c}}){e^c}_{{\tilde{b}}_j})){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
...rearrange...
$={e_u}^{\tilde{u}}({\partial }_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}})+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e^{a_i}}_{\tilde{c}}){e_{a_i}}^{{\tilde{a}}_i}+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_c}^{{\tilde{a}}_i}){e^c}_{\tilde{c}}+\underset{{\tilde{b}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_{b_i}}^{\tilde{c}}){e^{b_i}}_{{\tilde{b}}_i}-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_c}^{\tilde{c}}){e^c}_{{\tilde{b}}_j}){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
...using e times e-inverse equals identity, therefore ${\partial }_{\tilde{u}}({e^{a_i}}_{\tilde{c}}){e_{a_i}}^{{\tilde{a}}_i}=-{e^{a_i}}_{\tilde{c}}{\partial }_{\tilde{u}}({e_{a_i}}^{{\tilde{a}}_i})$
$={e_u}^{\tilde{u}}({\partial }_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}})+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}-\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_{a_i}}^{{\tilde{a}}_i}){e^{a_i}}_{\tilde{c}}+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_c}^{{\tilde{a}}_i}){e^c}_{\tilde{c}}+\underset{{\tilde{b}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_{b_i}}^{\tilde{c}}){e^{b_i}}_{{\tilde{b}}_i}-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {\partial }_{\tilde{u}}({e_c}^{\tilde{c}}){e^c}_{{\tilde{b}}_j}){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
...relabel sum indexes and cancel like terms:
$={e_u}^{\tilde{u}}({\partial }_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}})+\underset{{\tilde{a}}_i\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {{\tilde{\mathrm{\Gamma }}}^{{\tilde{a}}_i}}_{\tilde{u}\tilde{c}}-\underset{{\tilde{b}}_j\leftrightarrow \tilde{c}}{{T^{\tilde{A}}}_{\tilde{B}}}\cdot {{\tilde{\mathrm{\Gamma }}}^{\tilde{c}}}_{\tilde{u}{\tilde{b}}_j}){\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}}$
$={e_u}^{\tilde{u}}{\tilde{\mathrm{\nabla }}}_{\tilde{u}}({T^{\tilde{A}}}_{\tilde{B}}{\partial }_{\tilde{A}}\otimes d{x}^{\tilde{B}})$
Looks like the metric-cancelling torsion-free covariant-derivative of one basis is just the transform of the metric-cancelling torsion-free covariant-derivative of another.
Hence the covariant derivative of a tensor is a tensor.

---

Let the contorsion tensor be the difference between an arbitrary connection and the Levi-Civita connection:
${K^a}_{bc}={{\mathrm{\Gamma }}^a}_{bc}-{{\hat{\mathrm{\Gamma }}}^a}_{bc}$
$=\frac{1}{2}({{T_c}^a}_b+{{T_b}^a}_c+{T^a}_{bc})$

This shows that any connection can be uniquely defined by its contorsion, which is defined by its torsion.

Now let's revisit the definition of the non-coordinate metric-cancelling torsion-free connection with regards to the coordinate:
${{\hat{\mathrm{\Gamma }}}^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^a}_{bc}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})$
...substitute ${{\hat{\mathrm{\Gamma }}}^a}_{bc}={{\mathrm{\Gamma }}^a}_{bc}-{K^a}_{bc}$
${{\mathrm{\Gamma }}^a}_{bc}-{K^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^a}_{bc}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})$
...and solve:
${{\mathrm{\Gamma }}^a}_{bc}={{\tilde{\mathrm{\Gamma }}}^a}_{bc}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})+{K^a}_{bc}$
So any metric-cancelling connection can be represented as the sum of the coordinate-basis torsion-free connection, plus the partial of the vielbein, plus the contorsion.

And if there is one unique Levi-Civita torsion-free metric-cancelling connection per metric, and there is one unique linear transform from coordinate to non-coordinate basis per diagonalized metric, then there is only one unique torsion-free metric-cancelling connection per diagonalized metric (or per any non-coordinate basis?).
TODO but there isn't one unique transform for diagonalization.

Antisymmetry of contorsion tensor:
${K^a}_{[bc]}=\frac{1}{2}({{T_{[c}}^a}_{b]}+{{T_{[b}}^a}_{c]}+{T^a}_{[bc]})=\frac{1}{2}{T^a}_{[bc]}=\frac{1}{2}{T^a}_{bc}$

Trace of contorsion tensor:
${K^u}_{au}=\frac{1}{2}({{T_u}^u}_a+{{T_a}^u}_u+{T^u}_{au})$
...cancelling trace of the antisymmetric components of the torsion...
${K^u}_{au}=\frac{1}{2}({T^u}_{ua}+{T^u}_{au})$
...applying the antisymmetry of the torsion...
${K^u}_{au}=0$

Jacobi formula applied to metric tensor (more on worksheet #13):
$e_a(g)=g{g}^{uv}{e}_a(g_{uv})$
$\frac{1}{2g}{e}_a(g)=\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$
$\frac{1}{2}{e}_a(log|g|)=\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$
$e_a(log\sqrt{|g|})=\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$
...in a non-coordinate basis that is a linear map of a coordinate basis...
$e_a(log\sqrt{|g|})=\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$

Jacobi formula of a coordinate metric in a coordinate basis:
Remember, $g_{uv}=e_u\cdot e_v$ is a non-coordinate metric expressed in the non-coordinate basis and ${\tilde{g}}_{\tilde{u}\tilde{v}}={\partial }_{\tilde{u}}\cdot {\partial }_{\tilde{v}}$ is the coordinate metric expressed in the coordinate basis.
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{\tilde{g}}^{\tilde{u}\tilde{v}}{\partial }_{\tilde{a}}({\tilde{g}}_{\tilde{u}\tilde{v}})$
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{g}^{uv}{e_u}^{\tilde{u}}{e_v}^{\tilde{v}}{\partial }_{\tilde{a}}(g_{cd}{e^c}_{\tilde{u}}{e^d}_{\tilde{v}})$
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{g}^{uv}{e_u}^{\tilde{u}}{e_v}^{\tilde{v}}({\partial }_{\tilde{a}}(g_{cd}){e^c}_{\tilde{u}}{e^d}_{\tilde{v}}+g_{cd}{\partial }_{\tilde{a}}({e^c}_{\tilde{u}}){e^d}_{\tilde{v}}+g_{cd}{e^c}_{\tilde{u}}{\partial }_{\tilde{a}}({e^d}_{\tilde{v}}))$
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{g}^{uv}{\partial }_{\tilde{a}}(g_{uv})+{e_v}^{\tilde{u}}{\partial }_{\tilde{a}}({e^v}_{\tilde{u}})$
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{g}^{uv}{\partial }_{\tilde{a}}(g_{uv})-{e^v}_{\tilde{u}}{\partial }_{\tilde{a}}({e_v}^{\tilde{u}})$

Going further:
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{g}^{uv}{\partial }_{\tilde{a}}(g_{uv})+{e_v}^{\tilde{u}}{\partial }_{\tilde{a}}({e^v}_{\tilde{u}})$
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2g}{\partial }_{\tilde{a}}g+\frac{1}{e}{\partial }_{\tilde{a}}e$
${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})=\frac{1}{2}{\partial }_{\tilde{a}}(log(\sqrt{|g|}))+{\partial }_{\tilde{a}}(log(e))$
transform both sides:
${e_a}^{\tilde{a}}{\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})={e_a}^{\tilde{a}}{\partial }_{\tilde{a}}(log(\sqrt{|g|}))+{e_a}^{\tilde{a}}{\partial }_{\tilde{a}}(log(e))$
$e_a(log\sqrt{|\tilde{g}|})=e_a(log(\sqrt{|g|}))+e_a(log(e))$
$e_a(log\sqrt{|\tilde{g}|})={{\hat{\mathrm{\Gamma }}}^b}_{ab}+e_a(log(e))$

Trace of Levi-Civita connection on 1 and 3:
${{\hat{\mathrm{\Gamma }}}^u}_{au}=\frac{1}{2}{g}^{uv}(e_u(g_{va})+e_a(g_{vu})-e_v(g_{au})+c_{vau}+c_{vua}-c_{uav})$
${{\hat{\mathrm{\Gamma }}}^u}_{au}=\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$
${{\hat{\mathrm{\Gamma }}}^u}_{au}=e_a(log\sqrt{|g|})$
Mind you, even in non-coordinate orthonormal frames where $g=$ constant, $log(\sqrt{|g|})=$ constant, and $e(log(\sqrt{|g|}))=0$, this identity holds.
I mean, the trace of the connection isn't necessarily the volume element. Sometimes it is just zero.

Trace of Levi-Civita connection on 1 and 2:
${{\hat{\mathrm{\Gamma }}}^u}_{ua}=\frac{1}{2}{g}^{uv}(e_a(g_{vu})+e_u(g_{va})-e_v(g_{ua})+c_{vua}+c_{vau}-c_{auv})$
$=\frac{1}{2}{g}^{uv}{e}_a(g_{vu})-{c_{au}}^u$
$=e_a(log\sqrt{|g|})+{c_{ua}}^u$

using ${{\hat{\mathrm{\Gamma }}}^a}_{bc}={e^a}_{\tilde{a}}{e_b}^{\tilde{b}}{e_c}^{\tilde{c}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{a}}}_{\tilde{b}\tilde{c}}+{e^a}_{\tilde{a}}{e}_b({e_c}^{\tilde{a}})$
and using ${{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{a}\tilde{u}}=\frac{1}{\sqrt{|\tilde{g}|}}{\partial }_{\tilde{a}}(\sqrt{|\tilde{g}|})$
therefore
${{\hat{\mathrm{\Gamma }}}^u}_{au}={e_a}^{\tilde{a}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{a}\tilde{u}}+{e^u}_{\tilde{u}}{e}_a({e_u}^{\tilde{u}})$
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})={e_a}^{\tilde{a}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{a}\tilde{u}}-e_a({e^u}_{\tilde{u}}){e_u}^{\tilde{u}}$
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})=\frac{1}{\sqrt{|\tilde{g}|}}{e_a}^{\tilde{a}}{\partial }_{\tilde{a}}(\sqrt{|\tilde{g}|})-\frac{1}{e}{e}_a(e)$
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})=\frac{1}{\sqrt{|\tilde{g}|}}{e}_a(\sqrt{|\tilde{g}|})-\frac{1}{e}{e}_a(e)$
...and keep going to find...
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})=\frac{1}{e\sqrt{|g|}}{e}_a(e\sqrt{|g|})-\frac{1}{e}{e}_a(e)$
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})=\frac{1}{e\sqrt{|g|}}e{e}_a(\sqrt{|g|})+\frac{1}{e\sqrt{|g|}}\sqrt{|g|}{e}_a(e)-\frac{1}{e}{e}_a(e)$
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})=\frac{1}{\sqrt{g}}{e}_a(\sqrt{|g|})+\frac{1}{e}{e}_a(e)-\frac{1}{e}{e}_a(e)$
$\frac{1}{\sqrt{|g|}}{e}_a(\sqrt{|g|})=\frac{1}{\sqrt{g}}{e}_a(\sqrt{|g|})$

What about ${{\hat{\mathrm{\Gamma }}}^u}_{ua}={e_a}^{\tilde{a}}{{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{u}\tilde{a}}+{e^u}_{\tilde{u}}{e}_u({e_a}^{\tilde{u}})$
And ${{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{u}\tilde{a}}$ is symmetric and therefore equal to ${{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{a}\tilde{u}}$, which is equal to ${\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})$
${{\hat{\mathrm{\Gamma }}}^u}_{ua}={e_a}^{\tilde{a}}{\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})+{e^u}_{\tilde{u}}{e}_u({e_a}^{\tilde{u}})$
${{\hat{\mathrm{\Gamma }}}^u}_{ua}=e_a(log\sqrt{|\tilde{g}|})+{e^u}_{\tilde{u}}{e}_u({e_a}^{\tilde{u}})$

Equate the two definitions of ${{\hat{\mathrm{\Gamma }}}^u}_{ua}$ to find:
$e_a(log\sqrt{|g|})+{c_{ua}}^u=e_a(log\sqrt{|\tilde{g}|})+{e^u}_{\tilde{u}}{e}_u({e_a}^{\tilde{u}})$

Now this is the connection that a covariant divergence would use:
$\hat{\mathrm{\nabla }}\cdot v$
$=e^b{\hat{\mathrm{\nabla }}}_b(v^a{e}_a)$
$=e_b(v^a)e^b(e_a)+e^b({\mathrm{\nabla }}_b{e}_a)v^a$
$=e_a(v^a)+{{\hat{\mathrm{\Gamma }}}^b}_{ba}{v}^a$
$=e_a(v^a)+v^b{{\hat{\mathrm{\Gamma }}}^a}_{ab}$
$=e_a(v^a)+v^b(e_b(log(\sqrt{|\tilde{g}|}))+e_a({e_b}^{\tilde{b}}){e^a}_{\tilde{b}})$
$=e_a(v^a)+v^a\frac{1}{\sqrt{|\tilde{g}|}}{e}_a(\sqrt{|\tilde{g}|})+v^b{e}_a({e_b}^{\tilde{b}}){e^a}_{\tilde{b}}$
$=\frac{1}{\sqrt{|\tilde{g}|}}{e}_a(\sqrt{|\tilde{g}|}{v}^a)+v^b{e}_a({e_b}^{\tilde{b}}){e^a}_{\tilde{b}}$
$=\frac{1}{\sqrt{|\tilde{g}|}}{\partial }_{\tilde{a}}(\sqrt{|\tilde{g}|}{e_a}^{\tilde{a}}{v}^a)$
$=\frac{1}{\sqrt{|\tilde{g}|}}{e^b}_{\tilde{a}}{e}_b(\sqrt{|\tilde{g}|}{e_a}^{\tilde{a}}{v}^a)$
This is the Voss-Weyl identity.

Levi-Civita connection antisymmetry on 2nd and 3rd indexes:
${{\hat{\mathrm{\Gamma }}}^a}_{[bc]}$
$={{\mathrm{\Gamma }}^a}_{[bc]}-{K^a}_{[bc]}$
$=\frac{1}{2}({T^a}_{bc}+{c_{bc}}^a)-\frac{1}{2}{T^a}_{[bc]}$
$=\frac{1}{2}{c_{bc}}^a$
In a holonomic basis this gives ${{\hat{\mathrm{\Gamma }}}^a}_{[bc]}=0$

Notice that ${{\hat{\mathrm{\Gamma }}}^u}_{ua}={{\hat{\mathrm{\Gamma }}}^u}_{au}+{c_{ua}}^u$
Therefore ${{\hat{\mathrm{\Gamma }}}^u}_{ua}={{\hat{\mathrm{\Gamma }}}^u}_{au}+{c_{ua}}^u$
${{\hat{\mathrm{\Gamma }}}^u}_{ua}=e_a(log\sqrt{|g|})+{c_{ua}}^u$
Which looks just like our trace of the Levi-Civita connection above.

Trace of an arbitrary connection:
${{\mathrm{\Gamma }}^u}_{au}={{\hat{\mathrm{\Gamma }}}^u}_{au}+{K^u}_{au}=e_a(log\sqrt{|g|})$
${{\mathrm{\Gamma }}^u}_{ua}-{{\mathrm{\Gamma }}^u}_{au}={T^u}_{ua}+{c_{ua}}^u$
${{\mathrm{\Gamma }}^u}_{ua}=e_a(log\sqrt{|g|})+{c_{ua}}^u+{T^u}_{ua}$

Trace of the Levi-Civita connection of the coordinate basis (which will equal the volume gradient):
${{\tilde{\mathrm{\Gamma }}}^{\tilde{u}}}_{\tilde{a}\tilde{u}}={\partial }_{\tilde{a}}(log\sqrt{|\tilde{g}|})$
...transformed into non-coordinates...
${{\tilde{\mathrm{\Gamma }}}^u}_{au}=e_a(log\sqrt{|\tilde{g}|})$
Difference of Levi-Civita connection of the coordinate basis vs the non-coordinate basis:
${{\hat{\mathrm{\Gamma }}}^u}_{au}={{\tilde{\mathrm{\Gamma }}}^u}_{au}+{e^u}_{\tilde{u}}{e}_a({e_u}^{\tilde{u}})$
${{\tilde{\mathrm{\Gamma }}}^u}_{au}={{\hat{\mathrm{\Gamma }}}^u}_{au}-{e^u}_{\tilde{u}}{e}_a({e_u}^{\tilde{u}})$
Trace of the Levi-Civita connection of the coordinate basis, in terms of the non-coordinate Levi-Civita connection:
${{\tilde{\mathrm{\Gamma }}}^u}_{au}=e_a(log\sqrt{|\tilde{g}|})$
Equate:
${{\hat{\mathrm{\Gamma }}}^u}_{au}-{e^u}_{\tilde{u}}{e}_a({e_u}^{\tilde{u}})=e_a(log\sqrt{|\tilde{g}|})$
... subsitute $-{e^u}_{\tilde{u}}{e}_a({e_u}^{\tilde{u}})=e_a(log\sqrt{|\tilde{g}|})-\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$ ...
${{\hat{\mathrm{\Gamma }}}^u}_{au}+e_a(log\sqrt{|\tilde{g}|})-\frac{1}{2}{g}^{uv}{e}_a(g_{uv})=e_a(log\sqrt{|\tilde{g}|})$
${{\hat{\mathrm{\Gamma }}}^u}_{au}=\frac{1}{2}{g}^{uv}{e}_a(g_{uv})$

---

Levi-Civita change-of-basis:
$e_c(g_{ab})=\frac{\partial }{\partial x^c}(g_{ab})$
$e_{c^{\prime }}(g_{a^{\prime }{b}^{\prime }}^{\prime })=\frac{\partial }{\partial x^{c^{\prime }}}(g_{a^{\prime }{b}^{\prime }}^{\prime })$
$=\frac{\partial }{\partial x^{c^{\prime }}}(g_{ab}^{\prime }\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}})$
$=\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial }{\partial x^c}(g_{ab}^{\prime }\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}})$
$=(\frac{\partial }{\partial x^c}(g_{ab})\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}+g_{ab}(\frac{\partial }{\partial x^c}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial }{\partial x^c}(\frac{\partial x^b}{\partial x^{b^{\prime }}})))\frac{\partial x^c}{\partial x^{c^{\prime }}}$
$=\frac{\partial }{\partial x^c}(g_{ab})\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{ab}(\frac{\partial }{\partial x^c}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial }{\partial x^c}(\frac{\partial x^b}{\partial x^{b^{\prime }}}))\frac{\partial x^c}{\partial x^{c^{\prime }}}$
$=\frac{\partial }{\partial x^c}(g_{ab})\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{ab}(\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}}))$

change of basis of connection:
${\mathrm{\Gamma }}_{abc}=\frac{1}{2}(e_c(g_{ab})+e_b(g_{ac})-e_a(g_{bc}))$
${\mathrm{\Gamma }}_{a^{\prime }{b}^{\prime }{c}^{\prime }}^{\prime }=\frac{1}{2}(e_{c^{\prime }}(g_{a^{\prime }{b}^{\prime }}^{\prime })+e_{b^{\prime }}(g_{a^{\prime }{c}^{\prime }}^{\prime })-e_{a^{\prime }}(g_{b^{\prime }{c}^{\prime }}^{\prime }))$
$=\frac{1}{2}(\frac{\partial }{\partial x^c}(g_{ab})\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{ab}(\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}}))$$+\frac{\partial }{\partial x^b}(g_{ac})\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}+g_{ac}(\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}+\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}}))$$-\frac{\partial }{\partial x^a}(g_{cb})\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^a}{\partial x^{a^{\prime }}}-g_{cb}(\frac{\partial }{\partial x^{a^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial }{\partial x^{a^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})))$
$=\frac{1}{2}(\frac{\partial }{\partial x^c}(g_{ab})+\frac{\partial }{\partial x^b}(g_{ac})-\frac{\partial }{\partial x^a}(g_{cb}))\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^a}{\partial x^{a^{\prime }}}+\frac{1}{2}(g_{ab}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+g_{ab}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^a}{\partial x^{a^{\prime }}}+g_{ac}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{ac}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^a}{\partial x^{a^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{a^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{a^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}})$
$={\mathrm{\Gamma }}_{abc}\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^a}{\partial x^{a^{\prime }}}+\frac{1}{2}(g_{ab}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+g_{ab}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^a}{\partial x^{a^{\prime }}}+g_{ac}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^a}{\partial x^{a^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{ac}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^a}{\partial x^{a^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{a^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{a^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}})$

${{{\mathrm{\Gamma }}^{\prime }}^{a^{\prime }}}_{b^{\prime }{c}^{\prime }}=g^{a^{\prime }{d}^{\prime }}{{\mathrm{\Gamma }}^{\prime }}_{d^{\prime }{b}^{\prime }{c}^{\prime }}$
$=g^{a^{\prime }{d}^{\prime }}({\mathrm{\Gamma }}_{dbc}\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^d}{\partial x^{d^{\prime }}}+\frac{1}{2}(g_{db}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^d}{\partial x^{d^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+g_{db}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^d}{\partial x^{d^{\prime }}}+g_{dc}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^d}{\partial x^{d^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{dc}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^d}{\partial x^{d^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{d^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{d^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}))$
$=\frac{\partial x^{a^{\prime }}}{\partial x^a}\frac{\partial x^{d^{\prime }}}{\partial x^e}{g}^{ae}({\mathrm{\Gamma }}_{dbc}\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^d}{\partial x^{d^{\prime }}}+\frac{1}{2}(g_{db}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^d}{\partial x^{d^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}+g_{db}\frac{\partial }{\partial x^{c^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^d}{\partial x^{d^{\prime }}}+g_{dc}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^d}{\partial x^{d^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}+g_{dc}\frac{\partial }{\partial x^{b^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^d}{\partial x^{d^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{d^{\prime }}}(\frac{\partial x^c}{\partial x^{c^{\prime }}})\frac{\partial x^b}{\partial x^{b^{\prime }}}-g_{cb}\frac{\partial }{\partial x^{d^{\prime }}}(\frac{\partial x^b}{\partial x^{b^{\prime }}})\frac{\partial x^c}{\partial x^{c^{\prime }}}))$
$=\frac{\partial x^{a^{\prime }}}{\partial x^a}{{\mathrm{\Gamma }}^a}_{bc}\frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^b}{\partial x^{b^{\prime }}}+\frac{\partial x^{a^{\prime }}}{\partial x^a}\frac{{\partial }^2{x}^a}{\partial x^{b^{\prime }}\partial x^{c^{\prime }}}$

geodesics minimize distance along a trajectory (using a coordinate basis):
Minimize $S$ by solving for $\delta S=0$:
Let $S=\int \sqrt{g_{uv}\frac{d{x}^u}{d\tau }\frac{d{x}^v}{d\tau }}d\tau$
We can just as easily minimize $S\to S=\int g_{uv}\frac{d{x}^u}{d\tau }\frac{d{x}^v}{d\tau }d\tau$
Now solve for $\frac{\delta S}{\delta x^a}=0$
$S=\int g_{uv}{\dot{x}}^u{\dot{x}}^{v}d\tau$
$\delta S=\int (\delta g_{uv}{\dot{x}}^u{\dot{x}}^v+2g_{uv}\delta {\dot{x}}^u{\dot{x}}^v)d\tau$
Next we use integration by parts and assign the total integral to zero: $udv=uv-vdu$ for $uv=0$
Using $u=2g_{uv}{\dot{x}}^v$, $dv=\delta {\dot{x}}^u$, and therefore $v=\delta x^u$, $du=2({\dot{g}}_{uv}{\dot{x}}^v+g_{uv}{\ddot{x}}^v)d\tau$
$\delta S=\int (\delta g_{uv}{\dot{x}}^u{\dot{x}}^v-2\delta x^u({\dot{g}}_{uv}{\dot{x}}^v+g_{uv}{\ddot{x}}^v))d\tau$
Substitute $\delta g_{uv}=\frac{\partial g_{uv}}{\partial x^w}\cdot \delta x^w$ and $\frac{\partial g_{uv}}{\partial \tau }=\frac{\partial g_{uv}}{\partial x^w}\cdot \frac{\partial x^w}{\partial \tau }$
$\delta S=\int (\delta x^w\frac{\partial }{\partial x^w}{g}_{uv}{\dot{x}}^u{\dot{x}}^v-2\delta x^u(\frac{\partial }{\partial x^w}{g}_{uv}{\dot{x}}^w{\dot{x}}^v+g_{uv}{\ddot{x}}^v))d\tau$
Reindex and factor out $\delta x^u$
$\delta S=\int \delta x^u(\frac{\partial }{\partial x^u}{g}_{ab}{\dot{x}}^a{\dot{x}}^b-2\frac{\partial }{\partial x^a}{g}_{ub}{\dot{x}}^a{\dot{x}}^b-2g_{uv}{\ddot{x}}^v)d\tau$
Solve for $\delta S=0$
$0=\frac{\partial }{\partial x^u}{g}_{ab}{\dot{x}}^a{\dot{x}}^b-2\frac{\partial }{\partial x^a}{g}_{ub}{\dot{x}}^a{\dot{x}}^b-2g_{uv}{\ddot{x}}^v$
$0=g^{cu}(\frac{\partial }{\partial x^u}{g}_{ab}{\dot{x}}^a{\dot{x}}^b-2\frac{\partial }{\partial x^a}{g}_{ub}{\dot{x}}^a{\dot{x}}^b-2g_{uv}{\ddot{x}}^v)$
${\ddot{x}}^c-\frac{1}{2}{g}^{cu}(\frac{\partial }{\partial x^u}{g}_{ab}-2\frac{\partial }{\partial x^a}{g}_{ub}){\dot{x}}^a{\dot{x}}^b=0$
Separate the term with a factor of 2:
${\ddot{x}}^c+\frac{1}{2}{g}^{cu}(\frac{\partial }{\partial x^b}{g}_{ua}+\frac{\partial }{\partial x^a}{g}_{ub}-\frac{\partial }{\partial x^u}{g}_{ab}){\dot{x}}^a{\dot{x}}^b=0$
${\ddot{x}}^c+{{\mathrm{\Gamma }}^c}_{ab}{\dot{x}}^a{\dot{x}}^b=0$
For ${{\mathrm{\Gamma }}^c}_{ab}=\frac{1}{2}{g}^{cu}(\frac{\partial }{\partial x^b}{g}_{ua}+\frac{\partial }{\partial x^a}{g}_{ub}-\frac{\partial }{\partial x^u}{g}_{ab})$
This fits with the above definition of a metric-cancelling covariant derivative that has no torsion and no commutation.

acceleration of geodesic motion:
${\hat{\mathrm{\nabla }}}_{u}u=0$
$u^b{\hat{\mathrm{\nabla }}}_{b}u=0$
$u^b({\partial }_b(u^a)+{{\hat{\mathrm{\Gamma }}}^a}_{bc}{u}^c)=0$
$u^b{\partial }_b(u^a)+{{\hat{\mathrm{\Gamma }}}^a}_{bc}{u}^b{u}^c=0$
${\dot{u}}^a=-{{\hat{\mathrm{\Gamma }}}^a}_{bc}{u}^b{u}^c$

...in the presence of torsion:
${\dot{u}}^a=-({{\mathrm{\Gamma }}^a}_{bc}-{K^a}_{bc})u^b{u}^c$
${\dot{u}}^a=-({{\mathrm{\Gamma }}^a}_{bc}-\frac{1}{2}({{T_c}^a}_b+{{T_b}^a}_c+{T^a}_{bc}))u^b{u}^c$
${\dot{u}}^a=-{{\mathrm{\Gamma }}^a}_{bc}{u}^b{u}^c+{{T_b}^a}_c{u}^b{u}^c$
${\dot{u}}^a=-{{\mathrm{\Gamma }}^a}_{bc}{u}^b{u}^c+u_b{T^b}_{dc}{u}^c{g}^{da}$

change of basis of torsion-free holonomic geodesic:
$\frac{{\partial }^2{x}^a}{\partial {\tau }^2}+{{\mathrm{\Gamma }}^a}_{bc}\frac{\partial x^b}{\partial \tau }\frac{\partial x^c}{\partial \tau }=0$
$\frac{\partial }{\partial \tau }(\frac{\partial x^a}{\partial x^{a^{\prime }}}\frac{\partial x^{a^{\prime }}}{\partial \tau })+{{\mathrm{\Gamma }}^a}_{bc}\cdot \frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^{b^{\prime }}}{\partial \tau }\cdot \frac{\partial x^c}{\partial x^{c^{\prime }}}\frac{\partial x^{c^{\prime }}}{\partial \tau }=0$
$\frac{\partial }{\partial \tau }\frac{\partial x^a}{\partial x^{a^{\prime }}}\cdot \frac{\partial x^{a^{\prime }}}{\partial \tau }+\frac{\partial x^a}{\partial x^{a^{\prime }}}\cdot \frac{\partial }{\partial \tau }\frac{\partial x^{a^{\prime }}}{\partial \tau }+{{\mathrm{\Gamma }}^a}_{bc}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^c}{\partial x^{c^{\prime }}}\cdot \frac{\partial x^{b^{\prime }}}{\partial \tau }\frac{\partial x^{c^{\prime }}}{\partial \tau }=0$
$\frac{{\partial }^2{x}^{a^{\prime }}}{\partial {\tau }^2}+(\frac{\partial x^{a^{\prime }}}{\partial x^a}{{\mathrm{\Gamma }}^a}_{bc}\frac{\partial x^b}{\partial x^{b^{\prime }}}\frac{\partial x^c}{\partial x^{c^{\prime }}}+\frac{\partial x^{a^{\prime }}}{\partial x^a}\frac{{\partial }^2{x}^a}{\partial x^{b^{\prime }}\partial x^{c^{\prime }}})\frac{\partial x^{b^{\prime }}}{\partial \tau }\frac{\partial x^{c^{\prime }}}{\partial \tau }=0$
This matches the change-of-basis of metric-cancelling connection, which we can now substitute:
$\frac{{\partial }^2{x}^{a^{\prime }}}{\partial {\tau }^2}+{{\mathrm{\Gamma }}^{a^{\prime }}}_{b^{\prime }{c}^{\prime }}\frac{\partial x^{b^{\prime }}}{\partial \tau }\frac{\partial x^{c^{\prime }}}{\partial \tau }=0$

---

Holonomic Levi-Civita connection as the sum of two other connections, derived from their metrics:

$g_{ab}=g_{ab}^{\prime }+g_{ab}^{″}$
$g_{ab,c}=g_{ab,c}^{\prime }+g_{ab,c}^{″}$
Once again, (see Section 4 - metric tensor) ${r^a}_b=g^{\prime ac}{g}_{cb}^{″}$ and $r={r^a}_a$.

${\mathrm{\Gamma }}_{abc}$
$=\frac{1}{2}(g_{ab,c}+g_{ac,b}-g_{bc,a})$
$=\frac{1}{2}(g_{ab,c}^{\prime }+g_{ac,b}^{\prime }-g_{bc,a}^{\prime })+\frac{1}{2}(g_{ab,c}^{″}+g_{ac,b}^{″}-g_{bc,a}^{″})$
$={\mathrm{\Gamma }}_{abc}^{\prime }+{\mathrm{\Gamma }}_{abc}^{″}$

${{\mathrm{\Gamma }}^a}_{bc}$
$=g^{ad}{\mathrm{\Gamma }}_{dbc}$
$=({\mathrm{\Gamma }}_{dbc}^{\prime }+{\mathrm{\Gamma }}_{dbc}^{″})(g^{\prime ad}-(g^{\prime ae}{g}_{ef}^{″}{g}^{\prime fd})\cdot \frac{1}{1+r})$
$={\mathrm{\Gamma }}_{dbc}^{\prime }{g}^{\prime ad}-({\mathrm{\Gamma }}_{dbc}^{\prime }{g}^{\prime ae}{g}_{ef}^{″}{g}^{\prime fd})\cdot \frac{1}{1+r}+{\mathrm{\Gamma }}_{dbc}^{″}{g}^{\prime ad}-{\mathrm{\Gamma }}_{dbc}^{″}(g^{\prime ae}{g}_{ef}^{″}{g}^{\prime fd})\cdot \frac{1}{1+r}$
$={{\mathrm{\Gamma }}^{\prime a}}_{bc}+g^{\prime ad}{\mathrm{\Gamma }}_{dbc}^{″}-g^{\prime ae}{g}_{ef}^{″}{{\mathrm{\Gamma }}^{\prime f}}_{bc}\cdot \frac{1}{1+r}-g^{\prime ae}{g}_{ef}^{″}{g}^{\prime fd}{\mathrm{\Gamma }}_{dbc}^{″}\cdot \frac{1}{1+r}$
$={{\mathrm{\Gamma }}^{\prime a}}_{bc}+g^{\prime ad}{g}_{de}^{″}{g}^{″ef}{\mathrm{\Gamma }}_{fbc}^{″}-g^{\prime ae}{g}_{ef}^{″}{{\mathrm{\Gamma }}^{\prime f}}_{bc}\cdot \frac{1}{1+r}-g^{\prime ae}{g}_{ef}^{″}{g}^{\prime fd}{g}_{dg}^{″}{g}^{″gh}{\mathrm{\Gamma }}_{hbc}^{″}\cdot \frac{1}{1+r}$
$={{\mathrm{\Gamma }}^{\prime a}}_{bc}+g^{\prime ad}{g}_{de}^{″}{{\mathrm{\Gamma }}^{″e}}_{bc}-g^{\prime ae}{g}_{ef}^{″}{{\mathrm{\Gamma }}^{\prime f}}_{bc}\cdot \frac{1}{1+r}-g^{\prime ae}{g}_{ef}^{″}{g}^{\prime fd}{g}_{dg}^{″}{{\mathrm{\Gamma }}^{″g}}_{bc}\cdot \frac{1}{1+r}$
$={{\mathrm{\Gamma }}^{\prime a}}_{bc}+{r^a}_d{{\mathrm{\Gamma }}^{″d}}_{bc}-\frac{1}{1+r}{r^a}_d{{\mathrm{\Gamma }}^{\prime d}}_{bc}-\frac{1}{1+r}{r^a}_d{r^d}_e{{\mathrm{\Gamma }}^{″e}}_{bc}$

---

Tensor density.

Assuming $\mathrm{\nabla }$ is a metric-cancelling covariant derivative, calculate the covariant derivative of the scalar of the metric determinant:
${\mathrm{\nabla }}_{a}g$
$={\mathrm{\nabla }}_a\left({\widetilde{ϵ}}^{u_1...u_n}\underset{i=1}{\stackrel{n}{\mathrm{\Pi }}}{g}_{i{u}_i}\right)$
$={\widetilde{ϵ}}^{u_1...u_n}{\mathrm{\nabla }}_a\left(\underset{i=1}{\stackrel{n}{\mathrm{\Pi }}}{g}_{i{u}_i}\right)$
$={\widetilde{ϵ}}^{u_1...u_n}\underset{i=1}{\stackrel{n}{\mathrm{\Sigma }}}({\mathrm{\nabla }}_a{g}_{i{u}_i}\cdot \underset{j=1,j\ne i}{\stackrel{n}{\mathrm{\Pi }}}{g}_{j{u}_j})$
Using ${\mathrm{\nabla }}_a{g}_{uv}=0$:
$={\widetilde{ϵ}}^{u_1...u_n}\underset{i=1}{\stackrel{n}{\mathrm{\Sigma }}}(0\cdot \underset{j=1,j\ne i}{\stackrel{n}{\mathrm{\Pi }}}{g}_{j{u}_j})$
$={\widetilde{ϵ}}^{u_1...u_n}\underset{i=1}{\stackrel{n}{\mathrm{\Sigma }}}0$
$=0$

Now we see that the covariant derivative of every scalar is not necessarily equal to the partial derivative of that scalar.

Raising it to a power:
${\mathrm{\nabla }}_a(g{)}^m$
$=m(g{)}^{m-1}\cdot {\mathrm{\nabla }}_{a}g$
$=m(g{)}^{m-1}\cdot 0$
$=0$

Equating this to the original "covariant derivative equals partial derivative plus connection" rule:
${{\mathrm{\Gamma }}^u}_{au}=e_a(log(\sqrt{|g|}))=e_a(\frac{1}{2}log(|g|))=\frac{1}{2|g|}{e}_a(|g|)=\frac{1}{2g}{e}_a(g)$
So $2g{{\mathrm{\Gamma }}^u}_{au}=e_a(g)$
$0={\mathrm{\nabla }}_{a}g=e_a(g)-e_a(g)=e_a(g)-2g{{\mathrm{\Gamma }}^u}_{au}$

Same but for metric determinant raised to a power:
$0={\mathrm{\nabla }}_a(g{)}^m$
$=e_a((g{)}^m)-e_a((g{)}^m)$
$=e_a((g{)}^m)-m(g{)}^{m-1}{e}_a(g)$
$=e_a((g{)}^m)-m(g{)}^{m-1}\cdot 2g{{\mathrm{\Gamma }}^u}_{au}$
$=e_a((g{)}^m)-2m(g{)}^m{{\mathrm{\Gamma }}^u}_{au}$

Using the metric determinant square-root:
${\mathrm{\nabla }}_a(\sqrt{|g|}{)}^m=e_a((\sqrt{|g|}{)}^m)-m(\sqrt{|g|}{)}^m{{\mathrm{\Gamma }}^u}_{au}$
From here on we will refer to any scalar function mapping from our manifold to a real value, if it is scaled by $(\sqrt{|g|}{)}^m$, then it will be "of weight $m$".

Combining this with the original (p, q) tensor definition of covariant derivatives:
Let $T={T^{a_1...a_p}}_{b_1...b_q}\underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$ be a (p, q) tensor.
Let $\mathcal{T}=(\sqrt{|g|}{)}^{m}T={{\mathcal{T}}^{a_1...a_p}}_{b_1...b_q}\underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}=(\sqrt{|g|}{)}^m{T^{a_1...a_p}}_{b_1...b_q}\underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$ be a weight-m (p, q) tensor.

The covariant derivative of $\mathcal{T}$ is defined as:
$\mathrm{\nabla }\mathcal{T}=e^u\otimes {\mathrm{\nabla }}_u((\sqrt{|g|}{)}^m{T^{a_1...a_p}}_{b_1...b_q}\underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i})$
$=e^u\otimes {\mathrm{\nabla }}_u((\sqrt{|g|}{)}^m){T^{a_1...a_p}}_{b_1...b_q}\underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}+e^u\otimes (\sqrt{|g|}{)}^m{\mathrm{\nabla }}_u({T^{a_1...a_p}}_{b_1...b_q}\underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i})$
$=((e_u((\sqrt{|g|}{)}^m)-m(\sqrt{|g|}{)}^m{{\mathrm{\Gamma }}^v}_{uv}){T^{a_1...a_p}}_{b_1...b_q}+(\sqrt{|g|}{)}^m(e_u({T^{a_1...a_p}}_{b_1...b_q})+{{\mathrm{\Gamma }}^{a_i}}_{uv}{T^{a_1...a_{i-1}v{a}_{i+1}...a_p}}_{b_1...b_q}-{{\mathrm{\Gamma }}^v}_{u{b}_i}{T^{a_1...a_p}}_{b_1...b_{i-1}v{b}_{i+1}...b_q}))e^u\otimes \underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$
$=(e_u((\sqrt{|g|}{)}^m{T^{a_1...a_p}}_{b_1...b_q})-m(\sqrt{|g|}{)}^m{{\mathrm{\Gamma }}^v}_{uv}{T^{a_1...a_p}}_{b_1...b_q}+(\sqrt{|g|}{)}^m(+{{\mathrm{\Gamma }}^{a_i}}_{uv}{T^{a_1...a_{i-1}v{a}_{i+1}...a_p}}_{b_1...b_q}-{{\mathrm{\Gamma }}^v}_{u{b}_i}{T^{a_1...a_p}}_{b_1...b_{i-1}v{b}_{i+1}...b_q}))e^u\otimes \underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$
$=(e_u({{\mathcal{T}}^{a_1...a_p}}_{b_1...b_q})+{{\mathrm{\Gamma }}^{a_i}}_{uv}{{\mathcal{T}}^{a_1...a_{i-1}v{a}_{i+1}...a_p}}_{b_1...b_q}-{{\mathrm{\Gamma }}^v}_{u{b}_i}{{\mathcal{T}}^{a_1...a_p}}_{b_1...b_{i-1}v{b}_{i+1}...b_q}-m{{\mathrm{\Gamma }}^v}_{uv}{{\mathcal{T}}^{a_1...a_p}}_{b_1...b_q})e^u\otimes \underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$

And now we have the up-add/down-subtract definition for the metric-cancelling covariant derivative of a tensor density of weight $m$.
Keep twisting the knobs and we get...
$=(e_u({\sqrt{|g|}}^m{T^{a_1...a_p}}_{b_1...b_q})+{\sqrt{|g|}}^m{{\mathrm{\Gamma }}^{a_i}}_{uv}{T^{a_1...a_{i-1}v{a}_{i+1}...a_p}}_{b_1...b_q}-{\sqrt{|g|}}^m{{\mathrm{\Gamma }}^v}_{u{b}_i}{T^{a_1...a_p}}_{b_1...b_{i-1}v{b}_{i+1}...b_q}-e_u({\sqrt{|g|}}^m){T^{a_1...a_p}}_{b_1...b_q})e^u\otimes \underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$
$={\sqrt{|g|}}^m(e_u({T^{a_1...a_p}}_{b_1...b_q})+{{\mathrm{\Gamma }}^{a_i}}_{uv}{T^{a_1...a_{i-1}v{a}_{i+1}...a_p}}_{b_1...b_q}-{{\mathrm{\Gamma }}^v}_{u{b}_i}{T^{a_1...a_p}}_{b_1...b_{i-1}v{b}_{i+1}...b_q})e^u\otimes \underset{i=1}{\stackrel{p}{\otimes }}{e}_{a_i}\underset{i=1}{\stackrel{q}{\otimes }}{e}^{b_i}$
$={\sqrt{|g|}}^m\mathrm{\nabla }T$
And this is where I could have ended up much earlier if I had just used the chain rule without expanding the basis vectors/one-forms of the unweighted tensor's definition.

Back