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Coordinate system:

Equations:

coordinate system \( x_i \)
vector basis: \( \textbf{e}_j = {e_j}^{\hat k}\)
one-form basis: \( \boldsymbol{\omega}^i = {\omega^i}_{\hat k} \)
for indexes \(i\) in the coordinates and indexes \({\hat k}\) in the embedded space
contravariant representation \( {\textbf v} = v^i {\textbf e}_i \)
covariant representation \( \textbf{w} = w_i \boldsymbol{\omega}^i \)
orthogonality of vector and one-form basis: \( \langle \boldsymbol{\omega}^i, {\textbf e}_j \rangle = \delta^i_j \)
elements of vectors: \( \langle \boldsymbol{\omega}^i, \textbf{v} \rangle = \langle \boldsymbol{\omega}^i, v^j \textbf{e}_j \rangle = v^j \langle \boldsymbol{\omega}^i, \textbf{e}_j \rangle = v^j \delta^i_j = v^i \)
elements of one-forms: \( \langle \textbf{w}, \textbf{e}_i \rangle = \langle w_j \boldsymbol{\omega}^j, \textbf{e}_i \rangle = w_j \langle \boldsymbol{\omega}^j, \textbf{e}_i \rangle = w_j \delta^i_j = w_i \)

inner product of vector and one-form: \( \langle \textbf{w}, \textbf{v} \rangle = \langle w_i \boldsymbol{\omega}^i, v^j \textbf{e}_j \rangle = w_i v^j \langle \boldsymbol{\omega}^i, \textbf{e}_j \rangle = w_i v^j \delta^i_j = w_i v^i \)
inner product symmetry: \( \langle \textbf{a}, \textbf{b} \rangle = \langle \textbf{b}, \textbf{a} \rangle \)
dot product of vectors: \( \textbf{a} \cdot \textbf{b} = a^i \textbf{e}_i \cdot b^j \textbf{e}_j = a^i b^j \textbf{e}_i \cdot \textbf{e}_j = a^i b^j g_{ij} = a^i b_i = a_i b^i \)
dot product symmetry: \( \textbf{a} \cdot \textbf{b} = \textbf{b} \cdot \textbf{a} \)
outer product of vectors: \( \textbf{a} \otimes \textbf{b} = a^i \textbf{e}_i \otimes b^j \textbf{e}_j = a^i b^j \textbf{e}_i \otimes \textbf{e}_j \)
outer product of one-forms: \( \textbf{a} \otimes \textbf{b} = a_i \boldsymbol{\omega}^i \otimes b_j \boldsymbol{\omega}^j = a_i b_j \boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j \)
wedge product of one-form basis: \( \boldsymbol{\omega}^i \wedge \boldsymbol{\omega}^j = \boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j - \boldsymbol{\omega}^j \otimes \boldsymbol{\omega}^i \)
wedge product of one-forms: \( \textbf{a} \wedge \textbf{b} = a_i \boldsymbol{\omega}^i \wedge b_j \boldsymbol{\omega}^j = a_i b_j \boldsymbol{\omega}^i \wedge \boldsymbol{\omega}^j = a_i b_j (\boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j - \boldsymbol{\omega}^j \otimes \boldsymbol{\omega}^i) = (a_i b_j - b_i a_j) \boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j = \textbf{a} \otimes \textbf{b} - \textbf{b} \otimes \textbf{a} \)
wedge product basis antisymmetry: \( \textbf{dx}^i \wedge \textbf{dx}^j = -\textbf{dx}^j \wedge \textbf{dx}^i \)
wedge product antisymmetry: \( \textbf{a} \wedge \textbf{b} = (-1)^{pq} \textbf{b} \wedge \textbf{a} \) for p-form \(\textbf{a}\) and q-form \(\textbf{b}\).

commutation coefficients: \( [\textbf{e}_i, \textbf{e}_j] = \boldsymbol{\nabla}_{\textbf{e}_i} \textbf{e}_j - \boldsymbol{\nabla}_{\textbf{e}_j} \textbf{e}_i = \nabla_i \textbf{e}_j - \nabla_j \textbf{e}_i = {c_{ij}}^k \textbf{e}_k \)
commutation coefficient antisymmetry: \( [ \textbf{e}_i, \textbf{e}_j ] = -[ \textbf{e}_j, \textbf{e}_i ] \) implies \( {c_{ij}}^k = -{c_{ji}}^k \)

A basis is holonomic when all \( {c_{ij}}^k = 0 \).
This coincides with when \( \textbf{e}_i = {\partial \over {\partial x^i}} \) such that \( [\textbf{e}_i, \textbf{e}_j] = \left [ {\partial \over {\partial x^i}}, {\partial \over {\partial x^j}} \right ] = 0 \). Subsequently \( \boldsymbol{\omega}^i = \textbf{dx}^i \).
A basis that does not meet this condition is considered anholonomic.

metric: \( g_{ij} = {\textbf e}_i \cdot {\textbf e}_j \)
metric inverse: \( ||g^{ij}|| = \) inverse of \( ||g_{ij}|| \) such that \( g^{ik} g_{kj} = \delta^i_j \)
metric symmetry: \( g_{ij} = \textbf{e}_i \cdot \textbf{e}_j = \textbf{e}_j \cdot \textbf{e}_i = g_{ji} \)
metric inverse symmetry implied from metric symmetry.
identity derivative: \( 0 = \partial_k \delta^i_j = \partial_k (g^{il} g_{lj}) = \partial_k g^{il} g_{lj} + g^{il} \partial_k g_{lj} \)

connection coefficient of the 1st kind: \( \Gamma_{ijk} \)
in a holonmic basis: \( \Gamma_{ijk} = {1 \over 2} (\partial_k g_{ij} + \partial_j g_{ik} - \partial_i g_{jk}) \)
in an anholonomic basis: \( \Gamma_{ijk} = {1 \over 2} (\partial_k g_{ij} + \partial_j g_{ik} - \partial_i g_{jk} + c_{ijk} + c_{ikj} - c_{jki}) \)
1st kind symmetry: \( \Gamma_{i[jk]} = {1 \over 2} (\Gamma_{ijk} - \Gamma_{ikj}) = {1 \over 4} (\partial_k g_{ij} + \partial_j g_{ik} - \partial_i g_{jk} + c_{ijk} + c_{ikj} - c_{jki} - \partial_j g_{ik} - \partial_k g_{ij} + \partial_i g_{kj} - c_{ikj} - c_{ijk} + c_{kji}) = {1 \over 4} ( c_{jki} - c_{kji} ) = {1 \over 2} c_{kji} \)
which reduces to 0 in a holonomic basis
1st kind antisymmetry: \( \Gamma_{(ij)k} = {1 \over 2} (\Gamma_{ijk} + \Gamma_{jik}) = {1 \over 4} (\partial_k g_{ij} + \partial_j g_{ik} - \partial_i g_{jk} + c_{ijk} + c_{ikj} - c_{jki} + \partial_k g_{ji} + \partial_i g_{jk} - \partial_j g_{ik} + c_{jik} + c_{jki} - c_{ikj}) = {1 \over 4} (2 \partial_k g_{ij} + c_{ijk} + c_{jik}) = {1 \over 2} \partial_k g_{ij} \)

connection coefficient of the 2nd kind: \( {\Gamma^i}_{jk} = g^{il} \Gamma_{ljk} \)

covariant derivative: \( \boldsymbol{\nabla} \)
covariant derivative per component: \( \boldsymbol{\nabla} = \nabla_i \boldsymbol{\omega}^i \)
covariant derivative of scalar: \( \boldsymbol{\nabla} \phi = \nabla_i \phi \boldsymbol{\omega}^i = \partial_i \phi \boldsymbol{\omega}^i \)
covariant derivative of inner product: \( \boldsymbol{\nabla} \langle \textbf{a}, \textbf{b} \rangle = \langle \boldsymbol{\nabla} \textbf{a}, \textbf{b} \rangle + \langle \textbf{a}, \boldsymbol{\nabla} \textbf{b} \rangle \)
covariant derivative of dot product: \( \boldsymbol{\nabla} ( \textbf{a} \cdot \textbf{b} ) = (\boldsymbol{\nabla} \textbf{a}) \cdot \textbf{b} + \textbf{a} \cdot (\boldsymbol{\nabla} \textbf{b}) \)
covariant derivative of outer product: \( \boldsymbol{\nabla} ( \textbf{a} \otimes \textbf{b} ) = (\boldsymbol{\nabla} \textbf{a}) \otimes \textbf{b} + \textbf{a} \otimes (\boldsymbol{\nabla} \textbf{b}) \)

\( 0 = \nabla_k \delta^i_j = \nabla_k \langle \boldsymbol{\omega}^i, {\textbf e}_j \rangle = \langle \nabla_k \boldsymbol{\omega}^i, {\textbf e}_j \rangle + \langle \boldsymbol{\omega}^i, \nabla_k {\textbf e}_j \rangle \)
so \( \langle \nabla_k \boldsymbol{\omega}^i, {\textbf e}_j \rangle = - \langle \boldsymbol{\omega}^i, \nabla_k {\textbf e}_j \rangle \)
let \( \langle \boldsymbol{\omega}^i, \nabla_k {\textbf e}_j \rangle = {\Gamma^i}_{jk} \)
so \( \langle \nabla_k \boldsymbol{\omega}^i, {\textbf e}_j \rangle = -{\Gamma^i}_{jk} \)

covariant derivative of vector basis: \( \langle \nabla_k {\textbf e}_j, \boldsymbol{\omega}^i \rangle = {\Gamma^i}_{jk} \), so \( \nabla_k {\textbf e}_j = {\Gamma^i}_{jk} {\textbf e}_i \)
covariant derivative of one-form basis: \( -\langle \nabla_k \boldsymbol{\omega}^i, {\textbf e}_j \rangle = {\Gamma^i}_{jk} \), so \( \nabla_k \boldsymbol{\omega}^i = -{\Gamma^i}_{jk} \boldsymbol{\omega}^j \)

covariant derivative of vector: \( \boldsymbol{\nabla} \textbf{v} = \nabla_j \textbf{v} \otimes \boldsymbol{\omega}^j = \nabla_j (v^i \textbf{e}_i) \otimes \boldsymbol{\omega}^j = (\nabla_j v^i \textbf{e}_i + v^i \nabla_j \textbf{e}_i) \otimes \boldsymbol{\omega}^j = (\partial_j v^k + v^i {\Gamma^k}_{ji}) \textbf{e}_k \otimes \boldsymbol{\omega}^j \)
covariant derivative of vector in index notation: \( \nabla_i v^j = \partial_i v^j + {\Gamma^j}_{ki} v^k \)

covariant derivative of one-form: \( \boldsymbol{\nabla} \textbf{w} = \nabla_j \textbf{w} \otimes \boldsymbol{\omega}^j = \nabla_j (w_i \boldsymbol{\omega}^i) \otimes \boldsymbol{\omega}^j = (\nabla_j w_i \boldsymbol{\omega}^i + w_i \nabla_j \boldsymbol{\omega}^i) \otimes \boldsymbol{\omega}^j = (\partial_j w_k - {\Gamma^k}_{ji} w_i ) \boldsymbol{\omega}^k \otimes \boldsymbol{\omega}^j \)
covariant derivative of one-form in index notation: \( \nabla_i w_j = \partial_i w_j - {\Gamma^k}_{ji} w_k \)

This is where we see that in index notation \( \nabla_k \) treats \( v^i \) as a vector due to its raised index, while in object notation \( \nabla_k \) treats \( v^i \) as a scalars of the \( \textbf{e}_i \) vector.

covariant derivative of metric in index notation: \( \nabla_k g_{ij} = \partial_k g_{ij} - {\Gamma^l}_{ik} g_{lj} - {\Gamma^l}_{jk} g_{il} = \partial_k g_{ij} - \Gamma_{jik} - \Gamma_{ijk} = \partial_k g_{ij} - \partial_k g_{ij} = 0 \)

covariant vector derivative: \( \nabla_{\textbf{u}} \textbf{v} = \textbf{u} \cdot \boldsymbol{\nabla} \textbf{v} = u^i \textbf{e}_i \cdot \nabla_j v^k \textbf{e}_k \otimes \boldsymbol{\omega}^j = u^i \textbf{e}_i \cdot (\partial_j v^k + v^l {\Gamma^k}_{jl}) \textbf{e}_k \otimes \boldsymbol{\omega}^j = u^i (\partial_j v^k + v^l {\Gamma^k}_{jl}) \textbf{e}_i \cdot \textbf{e}_k \otimes \boldsymbol{\omega}^j = u^i (\partial_j v^k + v^l {\Gamma^k}_{jl}) g_{ik} \boldsymbol{\omega}^j = u_k (\partial_j v^k + v^i {\Gamma^k}_{ji}) \boldsymbol{\omega}^j \)

Riemann curvature in terms of covariant derivative:
\( {R^i}_{jkl} = \langle \boldsymbol{\omega}^i, (\left [ \nabla_k, \nabla_l \right ] - \nabla_{[k,l]}) \textbf{e}_j \rangle \)
\( = \langle \boldsymbol{\omega}^i, \nabla_k \nabla_l \textbf{e}_j - \nabla_l \nabla_k \textbf{e}_j - \nabla_{[k,l]} \textbf{e}_j \rangle \)
\( = \langle \boldsymbol{\omega}^i, \nabla_k ({\Gamma^m}_{jl} \textbf{e}_m) - \nabla_l ({\Gamma^m}_{jk} \textbf{e}_m) - {c_{kl}}^m \nabla_m \textbf{e}_j \rangle \)
\( = \langle \boldsymbol{\omega}^i, \nabla_k {\Gamma^n}_{jl} \textbf{e}_n + {\Gamma^n}_{jl} \nabla_k \textbf{e}_n - \nabla_l {\Gamma^n}_{jk} \textbf{e}_n - {\Gamma^n}_{jk} \nabla_l \textbf{e}_n - {c_{kl}}^m {\Gamma^n}_{jm} \textbf{e}_n \rangle \)
\( = \langle \boldsymbol{\omega}^i, \textbf{e}_n \rangle ( \partial_k {\Gamma^n}_{jl} + {\Gamma^m}_{jl} {\Gamma^n}_{mk} - \partial_l {\Gamma^n}_{jk} - {\Gamma^m}_{jk} {\Gamma^n}_{ml} - {\Gamma^n}_{jm} {c_{kl}}^m ) \)
\( = \partial_k {\Gamma^i}_{jl} - \partial_l {\Gamma^i}_{jk} + {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{ml} {\Gamma^m}_{jk} - {\Gamma^i}_{jm} {c_{kl}}^m \)

Riemann curvature in terms of covariant derivative, in index notation:
\( (\left [ \nabla_k, \nabla_l \right ] - \nabla_{[k,l]} ) v^i \)
\(= \nabla_k \nabla_l v^i - \nabla_l \nabla_k v^i - \nabla_{[k,l]} v^i \)
\(= \nabla_k (\partial_l v^i + {\Gamma^i}_{jl} v^j) - \nabla_l (\partial_k v^i + {\Gamma^i}_{jk} v^j) - {c_{kl}}^m \nabla_m v^i \)
\(= \nabla_k \partial_l v^i + \nabla_k {\Gamma^i}_{jl} v^j + {\Gamma^i}_{jl} \nabla_k v^j - \nabla_l \partial_k v^i - \nabla_l {\Gamma^i}_{jk} v^j - {\Gamma^i}_{jk} \nabla_l v^j - {c_{kl}}^m (\partial_m v^i + {\Gamma^i}_{jm} v^j) \)
\(= \partial_k \partial_l v^i - {\Gamma^m}_{lk} \partial_m v^i + {\Gamma^i}_{mk} \partial_l v^m + \nabla_k {\Gamma^i}_{jl} v^j + {\Gamma^i}_{jl} \partial_k v^j + {\Gamma^i}_{jl} {\Gamma^j}_{mk} v^m - \partial_l \partial_k v^i + {\Gamma^m}_{kl} \partial_m v^i - {\Gamma^i}_{ml} \partial_k v^m - \nabla_l {\Gamma^i}_{jk} v^j - {\Gamma^i}_{jk} \partial_l v^j - {\Gamma^i}_{jk} {\Gamma^j}_{ml} v^m - {c_{kl}}^m \partial_m v^i - {\Gamma^i}_{jm} {c_{kl}}^m v^j \)
\( = \partial_k \partial_l v^i - \partial_l \partial_k v^i - {\Gamma^m}_{lk} \partial_m v^i + {\Gamma^m}_{kl} \partial_m v^i + {\Gamma^i}_{mk} \partial_l v^m - {\Gamma^i}_{jk} \partial_l v^j + {\Gamma^i}_{jl} \partial_k v^j - {\Gamma^i}_{ml} \partial_k v^m + \nabla_k {\Gamma^i}_{jl} v^j - \nabla_l {\Gamma^i}_{jk} v^j + {\Gamma^i}_{jl} {\Gamma^j}_{mk} v^m - {\Gamma^i}_{jk} {\Gamma^j}_{ml} v^m - {\Gamma^i}_{jm} {c_{kl}}^m v^j - {c_{kl}}^m \partial_m v^i \)
\( = (\nabla_k {\Gamma^i}_{jl} - \nabla_l {\Gamma^i}_{jk} + {\Gamma^i}_{ml} {\Gamma^m}_{jk} - {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{jm} {c_{kl}}^m ) v^j \)
\( = (\partial_k {\Gamma^i}_{jl} - \partial_l {\Gamma^i}_{jk} + {\Gamma^i}_{ml} {\Gamma^m}_{jk} - {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{jm} {c_{kl}}^m ) v^j \)
let \( {R^i}_{jkl} = \partial_k {\Gamma^i}_{jl} - \partial_l {\Gamma^i}_{jk} + {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{ml} {\Gamma^m}_{jk} - {\Gamma^i}_{jm} {c_{kl}}^m \)
so \( (\left [ \nabla_k \nabla_l \right ] - \nabla_{[k,l]}) v^i = {R^i}_{jkl} v^j \)

Ricci curvature: \( R_{ij} = {R^k}_{ikj} \)
Gaussian curvature: \( R = {R^i}_i \)

exterior derivative: \( \textbf{d} \)
exchangeability of exterior and covariant derivative for 2-forms: \( \textbf{dF} = \boldsymbol{\nabla} \cdot \textbf{*F}, \textbf{d*F} = \boldsymbol{\nabla} \cdot \textbf{F} \)
exterior derivative of scalar: \( \textbf{d}\phi = \partial_i \phi \textbf{dx}^i = \partial_i \phi \boldsymbol{\omega}^i \)
exterior derivative of one-form: \( \textbf{dw} = \textbf{d}(w_j \textbf{dx}^j) = \textbf{d}w_j \wedge \textbf{dx}^j + w_j \textbf{d}^2\textbf{x}^j = \textbf{d}w_j \wedge \textbf{dx}^j = \partial_i w_j \textbf{dx}^i \wedge \textbf{dx}^j = (\partial_i w_j - \partial_j w_i) \textbf{dx}^{|i} \wedge \textbf{dx}^{j|} \)
exterior derivative of wedge product of p-form \(\textbf{a}\) and q-form \(\textbf{b}\): \( \textbf{d} ( \textbf{a} \wedge \textbf{b} ) = \textbf{da} \wedge \textbf{b} + (-1)^p \textbf{a} \wedge \textbf{db} \)
exterior derivative of vector as a linear relation to vector basis: \( \textbf{de}_i = \textbf{e}_k \otimes {\boldsymbol{\omega}^k}_i \)
equating covariant and exterior derivative for vector basis: \( \textbf{de}_i = \boldsymbol{\nabla} \textbf{e}_i = {\Gamma^k}_{ij} \textbf{e}_k \otimes \boldsymbol{\omega}^j = \textbf{e}_k \otimes {\boldsymbol{\omega}^k}_i \)
so \( {\boldsymbol{\omega}^k}_i = {\Gamma^k}_{ij} \boldsymbol{\omega}^j \) and \( \nabla_i \boldsymbol{\omega}^k = -{\boldsymbol{\omega}^k}_i \)
exterior derivative expanded in a contravariant basis: \( \textbf{dv} = \textbf{d} (v^i \textbf{e}_i) = \textbf{d}v^i \otimes \textbf{e}_i + v^i \textbf{de}_i = (\textbf{d}v^k + v^i {\boldsymbol{\omega}^k}_i) \otimes \textbf{e}_k = (\partial_j v^k + {\Gamma^k}_{ij} v^i) \boldsymbol{\omega}^j \otimes \textbf{e}_k \)
exterior derivative of inner product: \( \textbf{d}(\textbf{u} \cdot \textbf{v}) = \textbf{du} \cdot \textbf{v} + \textbf{u} \cdot \textbf{dv} \)
exterior derivative of metric tensor: \( \textbf{d}g_{ij} = (\textbf{de}_i \cdot \textbf{e}_j) = \textbf{de}_i \cdot \textbf{e}_j + \textbf{e}_i \cdot \textbf{de}_j = \textbf{e}_k \otimes {\boldsymbol{\omega}^k}_i \cdot \textbf{e}_j + \textbf{e}_i \cdot \textbf{e}_k \otimes {\boldsymbol{\omega}^k}_j = g_{jk} {\boldsymbol{\omega}^k}_i + g_{ik} {\boldsymbol{\omega}^k}_j = \boldsymbol{\omega}_{ji} + \boldsymbol{\omega}_{ij} \)

second exterior derivative of a one-form: \( \textbf{d}(\textbf{dw}) = 0 \)
second exterior derivative of vector: \( \textbf{d}^2 \textbf{v} = \textbf{d}^2 ( v^i \textbf{e}_i ) = \textbf{d} ( \textbf{d} v^i \textbf{e}_i + v^i \textbf{de}_i ) = \textbf{d} ( \textbf{d} v^i \textbf{e}_i + v^i \textbf{e}_j {\boldsymbol{\omega}^j}_i ) = \textbf{d}^2 v^i \textbf{e}_i + \textbf{d} v^i \wedge \textbf{d} \textbf{e}_i + \textbf{d} v^i \wedge \textbf{e}_j {\boldsymbol{\omega}^j}_i + v^i \textbf{de}_j \wedge {\boldsymbol{\omega}^j}_i + v^i \textbf{e}_j \textbf{d}{\boldsymbol{\omega}^j}_i = \textbf{d}^2 v^i \textbf{e}_i + 2 \textbf{d} v^i \wedge \textbf{e}_j {\boldsymbol{\omega}^j}_i + v^i \textbf{e}_k {\boldsymbol{\omega}^k}_j \wedge {\boldsymbol{\omega}^j}_i + v^i \textbf{e}_j \textbf{d}{\boldsymbol{\omega}^j}_i \)
...zero second exterior derivatives of one-forms... \( = v^i \textbf{e}_j {\boldsymbol{\omega}^j}_k \wedge {\boldsymbol{\omega}^k}_i + v^i \textbf{e}_j \textbf{d}{\boldsymbol{\omega}^j}_i \)
let \( {\textbf{R}^i}_j = {\boldsymbol{\omega}^i}_k \wedge {\boldsymbol{\omega}^k}_j + \textbf{d}{\boldsymbol{\omega}^i}_j \)
so \( \textbf{d}^2 \textbf{v} = \textbf{e}_i {\textbf{R}^i}_j v^j \)

Riemann metric two-form:
\( {\textbf{R}^i}_j = {\boldsymbol{\omega}^i}_m \wedge {\boldsymbol{\omega}^m}_j + \textbf{d}{\boldsymbol{\omega}^i}_j \)
\( = {\Gamma^i}_{mk} {\Gamma^m}_{jl} \boldsymbol{\omega}^k \wedge \boldsymbol{\omega}^l + \textbf{d} ( {\Gamma^i}_{jl} \boldsymbol{\omega}^l ) \)
\( = {\Gamma^i}_{mk} {\Gamma^m}_{jl} \boldsymbol{\omega}^k \wedge \boldsymbol{\omega}^l + \textbf{d} {\Gamma^i}_{jl} \boldsymbol{\omega}^l + {\Gamma^i}_{jl} \textbf{d} \boldsymbol{\omega}^l \)
\( = ( \partial_k {\Gamma^i}_{jl} + {\Gamma^i}_{mk} {\Gamma^m}_{jl} ) \boldsymbol{\omega}^k \wedge \boldsymbol{\omega}^l \)
\( = ( \partial_k {\Gamma^i}_{jl} - \partial_l {\Gamma^i}_{jk} + {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{ml} {\Gamma^m}_{jk} ) \boldsymbol{\omega}^{|k} \wedge \boldsymbol{\omega}^{l|} \)
\( = {R^i}_{jkl}\boldsymbol{\omega}^{|k} \wedge \boldsymbol{\omega}^{l|} \)
\( = {R^i}_{j|kl|}\boldsymbol{\omega}^k \wedge \boldsymbol{\omega}^l \)

\( 0 = \textbf{d} P = \textbf{d} (\textbf{e}_k \otimes \boldsymbol{\omega}^k) = \textbf{de}_k \wedge \boldsymbol{\omega}^k + \textbf{e}_k \otimes \textbf{d}\boldsymbol{\omega}^k = \textbf{e}_i \otimes {\boldsymbol{\omega}^i}_k \wedge \boldsymbol{\omega}^k + \textbf{e}_i \otimes \textbf{d} \boldsymbol{\omega}^i = \textbf{e}_i \otimes ({\boldsymbol{\omega}^i}_k \wedge \boldsymbol{\omega}^k + \textbf{d} \boldsymbol{\omega}^i) \)
so \( {\boldsymbol{\omega}^i}_k \wedge \boldsymbol{\omega}^k + \textbf{d} \boldsymbol{\omega}^i = 0 \)
so \( \textbf{d} \boldsymbol{\omega}^k = -{\boldsymbol{\omega}^k}_i \wedge \boldsymbol{\omega}^i = -{\Gamma^k}_{ji} \boldsymbol{\omega}^j \wedge \boldsymbol{\omega}^i = {\Gamma^k}_{ji} \boldsymbol{\omega}^i \wedge \boldsymbol{\omega}^j = ({\Gamma^k}_{ji} - {\Gamma^k}_{ij}) \boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j = {c_{ij}}^k \boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j \)
This coincides with, in a holonomic basis, when \( \boldsymbol{\omega}^i = \textbf{d}x^i, \textbf{d} \boldsymbol{\omega}^i = \textbf{d}^2\textbf{x}^i = 0 \)

Sources:
Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1970), Gravitation, New York: W.H. Freeman. ISBN 0-7167-0344-0
Flanders, Harley (1989), Differential Forms With Applications to the Physical Sciences, Mineola, New York: Dover Publications, ISBN 0-486-66169-5

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