Coordinate independence
We check that the curvature defined in the notation of Sect. 4 by 4.5 which can be stated as
$$\begin{aligned} {}^\varphi \,{\text{R}}^{k}_{\lambda \mu \nu } = \sum _{i\in I} \lambda _{i} \sum _{l} \left( \left( x^{i}_{k, l} \circ y^{i} \right) _{,\mu } y^{i}_{l, \nu \lambda }\right) _{[\mu \nu ]} + \left( \lambda _{i, \mu } {}^i\,{\Gamma }_{\nu \lambda }^{k}\right) _{[\mu \nu ]} + \sum _{\kappa } \left( {}^\varphi \,{\Gamma }^{k}_{\mu \kappa }{}^\varphi \,{\Gamma }^\kappa _{\nu \lambda }\right) _{[\mu \nu ]} \end{aligned}$$
(A.1)
is actually coordinate independent. The standard textbook calculation could be referred to if a third derivative of transition maps would exists.
Lemma A.1
Let M be a manifold, \( \varphi _{i}:U_{i} \rightarrow M \) be charts in a \({{\mathrm{C}}^{2}}\)-atlas of M, and \( \lambda _{i} \) be a corresponding \({{\mathrm{C}}^{1}}\)-partition of unity. For any two charts \( \varphi :U \rightarrow M \) and \( \varphi ' :U'\rightarrow M \) the expression defined by (4.5) coincide on \( \varphi (U) \cap \varphi '(U') \) as (3,1)-tensor, i.e.
$$\begin{aligned} \sum _{\lambda ', \mu ', \nu ', k'} \left( {}^{\varphi ^{\prime }}\,{R}^{k'}_{\lambda '\mu '\nu '} \circ x'\right) x'_{\lambda ', \lambda } x'_{\mu ', \mu } x'_{\nu ', \nu } \left( x_{k, k'} \circ x'\right) = {}^\varphi\, {\text{R}}^{k}_{\lambda \mu \nu }. \end{aligned}$$
Proof
We repeat all crucial definitions in the primed and non-primed versions
$$\begin{aligned} x= \varphi^{-1} \circ \varphi ',x'= \varphi '^{-1} \circ \varphi ,\\ x^{i}= \varphi ^{-1} \circ \varphi _{i},x'^{i}= \varphi '^{-1} \circ \varphi _{i},\\ y^{i}= \varphi _{i}^{-1} \circ \varphi ,y^{\prime i}= \varphi _{i}^{-1} \circ \varphi ', \\ {}^i{\varGamma }_{\mu \nu }^{k}= \sum _{l} \left( x^{i}_{k,l} \circ y^{i}\right) \left( y^{i}_{l,\mu \nu }\right) ,{}^i{\varGamma }_{\mu \nu }^{\prime k}= \sum _{l} \left( x^{\prime i}_{k,l} \circ y^{\prime i}\right) \left( y^{\prime i}_{l,\mu \nu }\right) ,\\ {}^\varphi {\varGamma }_{\mu \nu }^{k}= \sum _{i\in I} \lambda _i\,{}^i{\varGamma }_{\mu \nu }^{k},{}^{\varphi '}{\varGamma }_{\mu \nu }^{k}= \sum _{i\in I} \lambda _{i} {}^i{\varGamma }_{\mu \nu }^{\prime k}. \end{aligned}$$
defined on the suitable domains and the abuse of notation \(\lambda _{i} = \lambda _{i} \circ \varphi \). Additionally, we introduce the shorthand
$$\begin{aligned} (h_{\mu \mu '\nu \nu '})_{[\mu \nu ]'} :=h_{\mu \mu '\nu \nu '} - h_{\nu \nu '\mu \mu '} . \end{aligned}$$
Observe that in case \( h_{\mu \mu '\nu \nu '} = f_{\mu \mu '} \cdot g_{\nu \nu '} \)
$$\begin{aligned} \nonumber \sum \limits _{\mu '\nu '} \left( h_{\mu '\mu '\nu '\nu '}\right) _{[\mu \nu ]'}&= \left( \sum \limits _{\mu '} f_{\mu \mu '} \right) \left( \sum \limits _{\nu '} g_{\nu \nu '} \right) - \left( \sum \limits _{\nu '} f_{\nu \nu '} \right) \left( \sum \limits _{\mu '} g_{\mu \mu '} \right) \\&= \left( \sum \limits _{\mu '} f_{\mu \mu '} \sum \limits _{\nu '} g_{\nu \nu '} \right) _{[\mu \nu ]}. \end{aligned}$$
(A.2)
We have \( x' :=\varphi '^{-1} \circ \varphi = x'^{i} \circ y^{i} \) and \( x :=\varphi ^{-1} \circ \varphi ' = x^{i} \circ y'^{i} \) where defined. Observe that for the differential \( D\,{ id }:{\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{d} \) of the identity the following identity holds
$$\begin{aligned} 0&= (D\,{ id })_{,\lambda } = ({\text{D}}\, x\circ x' )_{,\lambda } = \left( \left( \sum \limits _{l} x_{k,l} \circ x' \cdot x'_{l,\nu } \right) _{k\nu } \right) _{,\lambda } \nonumber \\&= \sum \limits _{l} \left( (x_{k,l} \circ x')_{,\lambda } x'_{l,\nu } + (x_{k,l} \circ x') x'_{l,\nu \lambda } \right) _{k\nu }. \end{aligned}$$
(A.3)
Finally, we prove the lemma. Fix any \( \lambda , \mu , \nu , k \in \{1,\ldots , d\} \) and observe
$$\begin{aligned} \sum _{\lambda ', \mu ', \nu ', k'} \big ({}^{\varphi ^{\prime }}\,{R}^{k'}_{\lambda '\mu '\nu '} {}\circ x'\big ) x'_{\lambda ', \lambda } x'_{\mu ', \mu } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \end{aligned}$$
plug in definition (A.1) for \( {}^{\varphi ^{\prime }}\,{R}^{k'}_{\lambda '\mu '\nu '} \) and use the abuse of notation \(\lambda _{i, \mu '} = (\lambda _{i} \circ \varphi ')_{,\mu } \)
$$\begin{aligned}&= \sum _{i\in I} \lambda _{i} \sum _{\lambda ', k', l'} \sum _{\mu ', \nu '} \left( \left( \left( x^{\prime i}_{k', l'} \circ y^{\prime i} \right) _{,\mu '} \circ x'\right) x'_{\mu ', \mu } \left( y^{\prime i}_{l', \nu '\lambda '} \circ x'\right) x'_{\lambda ', \lambda } x'_{\nu ', \nu } \left( x_{k, k'} \circ x'\right) \right) _{[\mu \nu ]'}\\&\quad + \sum _{ i\in I } \sum _{\lambda ', k'} \sum _{\mu ', \nu '} \left( \lambda _{i, \mu '} x'_{\mu ', \mu } \left( {}^{i}\Gamma_{\nu ' \lambda '}' \circ x'\right) ^{k'} x'_{\lambda ', \lambda } x'_{\nu ', \nu } \left( x_{k, k'} \circ x'\right) \right) _{[\mu \nu ]'}\\&\quad + \sum _{\lambda ', k', \kappa '} \sum _{\mu ', \nu '} \left( {}^{\varphi'} \Gamma^{k'}_{\mu '\kappa '} {}^{\varphi '}\Gamma^{\kappa '}_{\nu '\lambda '} x'_{\lambda ', \lambda } x'_{\mu ', \mu } x'_{\nu ', \nu } \left( x_{k, k'} \circ x'\right) \right) _{[\mu \nu ]'} \end{aligned}$$
apply formula (A.2) to each summand \(\sum _{\mu ', \nu '}(\ldots )_{[\mu \nu ]'}\)
$$\begin{aligned}&= \sum _{ i\in I } \sum _{\lambda ', k', l'} \lambda _{i} \Bigl ( \textstyle \overbrace{\textstyle \sum \limits _{\mu '} \big (\big (x^{\prime i}_{k', l'} \circ y^{\prime i} \big )_{,\mu '} \circ x'\big ) x'_{\mu ', \mu } }^{\text {change of variables}} \cdot \sum \limits _{\nu '} \underbrace{ \big (y^{\prime i}_{l', \nu '\lambda '} \circ x'\big ) x'_{\lambda ', \lambda } }_{\text {change of variables}} x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \Bigr )_{[\mu \nu ]}\\&\quad + \sum _{ i\in I } \sum_{\lambda ', k'} \Bigl ( \underbrace{\textstyle \sum _{\mu '} \lambda _{i, \mu '} x'_{\mu ', \mu } }_{\text {change of variables}} \underbrace{\textstyle \sum _{\nu '} \big ({}^i\,{\Gamma }_{\nu ' \lambda '}' \circ x'\big )^{k'} x'_{\lambda ', \lambda } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) }_{\text {apply formula}\, (4.1)} \Bigr )_{[\mu \nu ]} \\&\quad + \sum _{\lambda ', k', \kappa '} \left( \textstyle \sum \limits _{\mu '} {}^{\varphi '}\,{\Gamma }^{k'}_{\mu '\kappa '} x'_{\mu ', \mu } \big (x_{k, k'} \circ x'\big ) \sum \limits _{\nu '} {}^{\varphi '}\,{\Gamma }^{\kappa '}_{\nu '\lambda '} x'_{\lambda ', \lambda } x'_{\nu ', \nu } \right) _{[\mu \nu ]} \\&= \sum _{ i\in I } \sum _{k', l'} \lambda _{i} \Bigl (\textstyle \big ( \underbrace{x^{\prime i}_{k', l'}}_{=\big (x'\circ x^{i}\big )_{k',l'} } {}\circ {} \underbrace{y^{\prime i} \circ x'}_{= y^{i}} \Big )_{,\mu } \sum _{\nu '} \Big ( \underbrace{y^{\prime i}_{l', \nu '} }_{=(y^{i} \circ x)_{l',\nu '}} {}\circ x' \Big )_{, \lambda } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \Bigr )_{[\mu \nu ]} \\&\quad + \sum _{i\in I} \left( \ \lambda _{i, \mu } \left( {}^i\,{\Gamma }_{\nu \lambda }^{k} - \sum _{l} \big (x_{k,l} \circ x'\big ) \big (x'_{l,\nu \lambda }\big ) \right) \right) _{[\mu \nu ]} \\&\quad + \sum _{\lambda ', k', \kappa ', \mu ', \nu ', m} \Bigl ( {}^{\varphi '}\,{\Gamma }^{k'}_{\mu 'm} x'_{\mu ', \mu } (x_{k, k'} \circ x') \cdot \underbrace{ \delta _{m\kappa '} }_{ (*)} \cdot {}^{\varphi '}\,{\Gamma }^{\kappa '}_{\nu '\lambda '} x'_{\lambda ', \lambda } x'_{\nu ', \nu } \Bigr )_{[\mu \nu ]} \end{aligned}$$
\((*)\): \( \delta _{m\kappa '} = \delta _{m\kappa '} \circ x'\) = \((D \,{ id } )_{m\kappa '} \circ x' = (D (x'\circ x))_{m\kappa '} \circ x'\) = \( (\sum\nolimits_{ \kappa } x'_{m, \kappa } \circ x \cdot x_{\kappa , \kappa '} ) \circ x'\) = \( \sum\nolimits_\kappa x'_{m, \kappa } (x_{\kappa , \kappa '} \circ x')\)
$$\begin{aligned}&=\sum _{\begin{array}{c} i\in I \\ k', l' \end{array}} \lambda _{i} \left( \begin{aligned}&\textstyle \bigl (\bigl ( \sum _\kappa x'_{k', \kappa } \circ x^{i} \cdot x^{i}_{\kappa ,l'} \bigr ) \circ y^{i} \bigr ){}_{,\mu } \\&\textstyle \cdot \sum _{\nu '} \bigl (\bigl ( \sum _{l} y^{i}_{l', l}\circ x \cdot x_{l, \nu '}\bigr ) \circ x'\bigr ){}_{, \lambda } \textstyle \cdot x'_{\nu ', \nu } (x_{k, k'} \circ x') \end{aligned} \right) _{[\mu \nu ]} \\&\quad + \sum _{i\in I} \bigl ( \lambda _{i, \mu } {}^i\,{\Gamma }_{\nu \lambda }^{k} \bigr )_{[\mu \nu ]} - \sum _{l} \left( \left( \sum _{i\in I} \lambda _{i}\right) _{, \mu } \big (x_{k,l} \circ x'\big ) \big (x'_{l,\nu \lambda }\big ) \right) _{[\mu \nu ]} \\&\quad + \sum _\kappa \Biggl ( \underbrace{\textstyle \sum \limits _{k', \mu ', m} {}^{\varphi '}\,{\Gamma }^{k'}_{\mu 'm} x'_{\mu ', \mu } \big (x_{k, k'} \circ x'\big ) x'_{m, \kappa } }_{\text{apply formula }\, (4.1)} \underbrace{\textstyle \sum \limits _{\lambda ', \kappa ', \nu '} {}^{\varphi '}\,{\Gamma }^{\kappa '}_{\nu '\lambda '} x'_{\lambda ', \lambda } x'_{\nu ', \nu } (x_{\kappa , \kappa '} \circ x') }_{\text{apply formula }\, (4.1)} \Biggr )_{[\mu \nu ]} \\&= \sum _{\begin{array}{c} i\in I \\ k', l' \end{array}} \lambda _{i} \left( \sum _{\kappa } \big ( x'_{k', \kappa } \cdot x^{i}_{\kappa ,l'} \circ y^{i} \big )_{,\mu } \sum _{\nu , l} \big ( y^{i}_{l', l} \cdot x_{l, \nu '} \circ x'\big )_{, \lambda } x'_{\nu ', \nu } (x_{k, k'} \circ x') \right) _{[\mu \nu ]} \\& \quad + \smash {\overbrace{\sum _{i\in I} \bigl ( \lambda _{i, \mu } {}^i\,{\Gamma }_{\nu \lambda }^{k}\bigr )_{[\mu \nu ]}}^{=:M}} - \sum _{l} \Bigl ( \smash
{\overbrace{1_{, \mu } }^{=0}} (x_{k,l} \circ x') x'_{l,\nu \lambda } \Bigr )_{[\mu \nu ]} \\& \quad + \sum _{ \kappa } \left( \left( {}^{\varphi} \,{\Gamma }^{k}_{\mu \kappa } - \sum _{l} \big (x_{k,l} \circ x'\big ) x'_{l,\mu \kappa } \right) \left({}^ \varphi \,{\Gamma }^\kappa _{\nu \lambda } - \sum _{l} \big (x_{\kappa ,l} \circ
x'\big )
x'_{l, \nu \lambda } \right) \right) _{[\mu \nu ]} \end{aligned}$$
$$\begin{aligned}&= \sum _{\begin{array}{c} i\in I \\ k', l' \end{array}} \lambda _{i} \left( \textstyle \begin{aligned}&\textstyle \sum _{\kappa } \bigl ( x'_{k', \kappa \mu } \big (x^{i}_{\kappa ,l'} \circ y^{i}\big ) + x'_{k', \kappa } \big (x^{i}_{\kappa ,l'} \circ y^{i}\big )_{,\mu } \bigr ) \\&\textstyle \cdot \sum _{\nu ', l} \bigl ( y^{i}_{l', l\lambda } \big (x_{l, \nu '} \circ x'\big ) + y^{i}_{l', l } \big (x_{l, \nu '} \circ x'\big )_{, \lambda } \bigr ) x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \end{aligned} \right) _{[\mu \nu ]} \\& \quad + M + \sum _{\kappa } \textstyle \begin{aligned}&\textstyle \Big ( {}^\varphi\, {\Gamma }^{k}_{\mu \kappa } {}^\varphi\, {\Gamma }^\kappa _{\nu \lambda } + \sum _{l, l'} {}^\varphi \,{\Gamma }^{k}_{\mu \kappa } \big (x_{\kappa ,l} \circ x'\big ) x'_{l, \nu \lambda } \\&\textstyle - {}^\varphi \,{\Gamma }^\kappa _{\nu \lambda } \big (x_{k,l} \circ x'\big ) x'_{l,\mu \kappa } + \big (x_{k,l} \circ x'\big ) x'_{l,\mu \kappa } \smash {\underbrace{\big (x_{\kappa ,l'} \circ x'\big ) x'_{l', \nu \lambda }}_{ \text{use formula }\, (A.3) }} \Big )_{[\mu \nu ]} \end{aligned} \\&= \sum _{\begin{array}{c} i\in I \\ l, \kappa \end{array}} \lambda _{i} \left( \textstyle \begin{aligned}&\textstyle \sum _{k', l'} \bigl ( x'_{k', \kappa \mu } (x^{i}_{\kappa ,l'} \circ y^{i}) + x'_{k', \kappa } \big (x^{i}_{\kappa ,l'} \circ y^{i}\big )_{,\mu } \bigr ) \\&\textstyle \cdot \sum _{\nu } \bigl ( y^{i}_{l', l\lambda } \big (x_{l, \nu '} \circ x'\big ) + y^{i}_{l', l } \big (x_{l, \nu '} \circ x'\big )_{, \lambda } \bigr ) x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \end{aligned} \right) _{[\mu \nu ]} \\& \quad + \underbrace{ \begin{aligned}&{\textstyle M + \sum _{\kappa } {}^\varphi \,{\Gamma }^{k}_{\mu \kappa } {}^\varphi \,{\Gamma }^\kappa _{\nu \lambda } -\sum _{\kappa , l, l'} {}^\varphi \,{\Gamma }^{k}_{\mu \kappa } \big (x_{\kappa ,l} \circ x'\big ) x'_{l, \nu \lambda } } \\&+ {}^\varphi \,{\Gamma }^\kappa _{\nu \lambda } (x_{k,l} \circ x') x'_{l,\mu \kappa } + (x_{k,l} \circ x') x'_{l,\mu \kappa } (x_{\kappa ,l'} \circ x')_{,\lambda } x'_{l', \nu } \end{aligned} }_{=:T } \end{aligned}$$
$$\begin{aligned}&= \sum _{\begin{array}{c} i\in I,\\ l, \kappa \end{array}} \lambda _{i} \left( \begin{aligned}&\textstyle \sum \limits _{k'} x'_{k', \kappa \mu } \big (x_{k, k'} \circ x'\big ) \sum \limits _{l'} \big (x^{i}_{\kappa ,l'} \circ y^{i}\big ) y^{i}_{l', l\lambda } \sum \limits _{\nu '} \big (x_{l, \nu '} \circ x'\big ) x'_{\nu ', \nu } \\&+ \textstyle \sum \limits _{l'} \big (x^{i}_{\kappa ,l'} \circ y^{i}\big )_{,\mu }y^{i}_{l', l\lambda } \sum \limits _{k'} x'_{k', \kappa } \big (x_{k, k'} \circ x'\big ) \sum \limits _{\nu '} \big (x_{l, \nu '} \circ x'\big ) x'_{\nu ', \nu } \\&\textstyle + \sum \limits _{k', \nu '} x'_{k', \kappa \mu } \big (x_{l, \nu '} \circ x'\big )_{, \lambda } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \sum \limits _{l'} \big (x^{i}_{\kappa ,l'} \circ y^{i}\big ) y^{i}_{l', l } \\&\textstyle + \sum \limits _{l',\nu '} \smash {\underbrace{\big (x^{i}_{\kappa ,l'} \circ y^{i}\big )_{,\mu } y^{i}_{l', l }}_{ \text{use formula}\, (A.3)}} \smash {\underbrace{(x_{l, \nu '} \circ x')_{, \lambda } x'_{\nu ', \nu }}_{ \text{use formula }\, (A.3)} \sum \limits _{k'} (x_{k, k'} \circ x') x'_{k', \kappa }} \end{aligned} \right) _{[\mu \nu ]} + T\\&= \sum _{\begin{array}{c} i\in I \\ l, \kappa \end{array}} \lambda _{i} \left( \textstyle \begin{aligned}&\textstyle \sum _{k'} x'_{k', \kappa \mu } \big (x_{k, k'} \circ x'\big ) {}^i\,\Gamma ^\kappa _{l\lambda } \delta _{l\nu } \textstyle + \sum _{l'} \big (x^{i}_{\kappa ,l'} \circ y^{i}\big )_{,\mu }y^{i}_{l', l\lambda } \delta _{\kappa k} \delta _{l\nu } \\&\textstyle + \sum _{k', \nu '} x'_{k', \kappa \mu } \big (x_{l, \nu '} \circ x'\big )_{, \lambda } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \delta _{\kappa l} \\&\textstyle + \sum _{l',\nu '} \big (x^{i}_{\kappa ,l'} \circ y^{i}\big ) y^{i}_{l', l \mu } \big (x_{l, \nu '} \circ x'\big ) x'_{\nu ', \nu \lambda } \delta _{k \kappa } \end{aligned} \right) _{[\mu \nu ]} + T \\&= \sum _{\begin{array}{c} i\in I \\ l, \kappa \end{array}} \lambda _{i} \left( \textstyle \begin{aligned}&\textstyle \sum _{k'} x'_{k', \kappa \mu } \big (x_{k, k'} \circ x'\big ) {}^i\,\Gamma ^\kappa _{\nu \lambda } \textstyle + \sum _{l'} \big (x^{i}_{k ,l'} \circ y^{i}\big )_{,\mu }y^{i}_{l', \nu \lambda } \\&\textstyle + \sum _{k', \nu '} x'_{k', \kappa \mu } \big (x_{\kappa , \nu '} \circ x'\big )_{, \lambda } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) \\&\textstyle + \sum _{\nu '} {}^i\,\Gamma ^{k}_{\mu l} \big (x_{l, \nu '} \circ x'\big ) x'_{\nu ', \nu \lambda } \end{aligned} \right) _{[\mu \nu ]} + T\end{aligned}$$
$$\begin{aligned}& =\sum _{i\in I} \lambda _{i}\sum _{l, \kappa } \left( \textstyle \begin{aligned}&\textstyle \sum \limits _{k'} x'_{k', \kappa \mu } \big (x_{k, k'} \circ x'\big ) {}^\varphi \,\Gamma ^\kappa _{\nu \lambda } \textstyle + \sum \limits _{l'} \big (x^{i}_{k ,l'} \circ y^{i}\big )_{,\mu }y^{i}_{l', \nu \lambda } \\&\textstyle + \sum \limits _{k', \nu '} x'_{k', \kappa \mu } \big (x_{\kappa , \nu '} \circ x'\big )_{, \lambda } x'_{\nu ', \nu } \big (x_{k, k'} \circ x'\big ) + \sum \limits _{\nu '} {}^\varphi\, \Gamma ^{k}_{\mu l} \big (x_{l, \nu '} \circ x'\big ) x'_{\nu ', \nu \lambda } \end{aligned} \right) _{[\mu \nu ]} \\&+ \sum _{i\in I} \bigl ( \lambda _{i, \mu }{}^ i\,{\Gamma }_{\nu \lambda }^{k} \bigr )_{[\mu \nu ]} + \sum _{\kappa } {}^\varphi \,{\Gamma }^{k}_{\mu \kappa } {}^\varphi \,{\Gamma }^\kappa _{\nu \lambda } \\&- \sum _{\kappa , l, l'} {}^\varphi \,{\Gamma }^{k}_{\mu \kappa } \big (x_{\kappa ,l} \circ x'\big ) x'_{l, \nu \lambda } + {}^\varphi \,{\Gamma }^\kappa _{\nu \lambda } \big (x_{k,l} \circ x'\big ) x'_{l,\mu \kappa } + \big (x_{k,l} \circ x'\big ) x'_{l,\mu \kappa } \big (x_{\kappa ,l'} \circ x'\big )_{,\lambda } x'_{l', \nu } \\&= \sum _{i\in I} \lambda _{i} \sum _{l'} \bigl ( \big (x^{i}_{k ,l'} \circ y^{i}\big )_{,\mu }y^{i}_{l', \nu \lambda } \bigr )_{[\mu \nu ]} + \bigl (\lambda _{i, \mu }{}^ i\,{\Gamma }_{\nu \lambda }^{k}\bigr )_{[\mu \nu ]} + \sum _{ \kappa } \bigl ({}^\varphi \,{\Gamma }^{k}_{\mu \kappa } {}^\varphi \,{\Gamma }^\kappa _{\nu \lambda }\bigr )_{[\mu \nu ]} \\&= {}^\varphi \,{\text{R}}^{k}_{\lambda \mu \nu }. \end{aligned}$$
