\begin{gather*}
\intertext
{\texttt
{... use J to implement ...
\newline
...
\newline
...
\newline
... generate a tensor $S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}$ ...
\newline
... where ...
\newline
... $S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}$ has the characteristics of a torsion tensor ...
\newline
... (i.e. skew-symmetric with respect to $\beta$ and $\gamma$) ...
\newline
...
\newline
... to enable covariant differentiation of tensors ...
\newline
... use the transformation rule for the connection ...
\newline
...
$
\Gamma_{\beta\gamma}^{\,\alpha}
=
\Bigl\{
\begin{matrix}
\alpha \\
\beta\gamma
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}
$
...
}
}
\begin{align*}
\begin{split}
\Bigl\{
\begin{matrix}
\nu \\
\lambda\mu
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu}
=
&
+
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial y^\beta}
{\partial x^\lambda}
\frac
{\partial y^\gamma}
{\partial x^\mu}
\Biggl(
\Bigl\{
\begin{matrix}
\alpha \\
\beta\gamma
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}
\Biggr)
+
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial^2 y^\alpha}
{\partial x^\lambda\partial x^\mu}
\end{split}    \\
\intertext
{\texttt
{... so ...
}
}
\begin{split}
\Gamma_{\lambda\mu}^{\,\nu}
-
\Gamma_{\mu\lambda}^{\,\nu}
=
&
+
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial y^\beta}
{\partial x^\lambda}
\frac
{\partial y^\gamma}
{\partial x^\mu}
\Biggl(
\Bigl\{
\begin{matrix}
\alpha \\
\beta\gamma
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}
\Biggr)
+
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial^2 y^\alpha}
{\partial x^\lambda\partial x^\mu}    \\
&
-
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial y^\gamma}
{\partial x^\mu}
\frac
{\partial y^\beta}
{\partial x^\lambda}
\Biggl(
\Bigl\{
\begin{matrix}
\alpha \\
\gamma\beta
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\gamma\beta\,.}^{\,\,\,\,\,\,\,\,\alpha}
\Biggr)
-
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial^2 y^\alpha}
{\partial x^\mu\partial x^\lambda}    \\
S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\,\nu}
=
&
+
\frac
{\partial x^\nu}
{\partial y^\alpha}
\frac
{\partial y^\beta}
{\partial x^\lambda}
\frac
{\partial y^\gamma}
{\partial x^\mu}
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}
\end{split}    \\
\intertext
{\texttt
{... and ...
}
}
\begin{split}
&
+
\Bigl(
B_{\,.\,\alpha\beta\gamma,\lambda}^{\epsilon}
+
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\rho}\,
B_{\,.\,\alpha\lambda\rho}^{\epsilon}
\Bigr)    \\
&
+
\Bigl(
B_{\,.\,\alpha\gamma\lambda,\beta}^{\epsilon}
+
S_{\gamma\lambda\,.}^{\,\,\,\,\,\,\,\rho}\,
B_{\,.\,\alpha\beta\rho}^{\epsilon}
\Bigr)    \\
&
+
\Bigl(
B_{\,.\,\alpha\lambda\beta,\gamma}^{\epsilon}
+
S_{\lambda\beta\,.}^{\,\,\,\,\,\,\,\rho}\,
B_{\,.\,\alpha\gamma\rho}^{\epsilon}
\Bigr)
=
0
\end{split}
\end{align*}
\end{gather*}



\begin{gather*}
\intertext
{\texttt
{...
\newline
...
\newline
... derive the covariant derivative of a tensor $C_{\lambda}$ $(C_{\lambda,\mu})$ ...
\newline
...
\newline
... the connection ...
}
}
\begin{align*}
\begin{split}
\frac
{\partial^2 y^\epsilon}
{\partial x^\lambda\partial x^\mu}
=
&
+
\frac
{\partial y^\epsilon}
{\partial x^\nu}
\Biggl(
\Bigl\{
\begin{matrix}
\nu \\
\lambda\mu
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu}
\Biggr)
-
\frac
{\partial y^\beta}
{\partial x^\lambda}
\frac
{\partial y^\gamma}
{\partial x^\mu}
\Biggl(
\Bigl\{
\begin{matrix}
\epsilon \\
\beta\gamma
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\epsilon}
\Biggr)
\end{split}    \\
\intertext
{\texttt
{...
}
}
\begin{split}
C_\lambda
=
&
+
\frac{\partial y^\epsilon}{\partial x^\lambda}
A_\epsilon
\end{split}    \\
\begin{split}
\frac{\partial C_\lambda}{\partial x^\mu}
=
&
+
\frac{\partial y^\epsilon}{\partial x^\lambda}
\frac{\partial y^\gamma}{\partial x^\mu}
\frac{\partial A_\epsilon}{\partial y^\gamma}    \\
&
+
\frac{\partial^2 y^\epsilon}{\partial x^\lambda\partial x^\mu}
A_\epsilon
\end{split}    \\
\begin{split}
=
&
+
\frac{\partial y^\epsilon}{\partial x^\lambda}
\frac{\partial y^\gamma}{\partial x^\mu}
\frac{\partial A_\epsilon}{\partial y^\gamma}    \\
&
+
\frac
{\partial y^\epsilon}
{\partial x^\nu}
\Biggl(
\Bigl\{
\begin{matrix}
\nu \\
\lambda\mu
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu}
\Biggr)
A_\epsilon    \\
&
-
\frac
{\partial y^\beta}
{\partial x^\lambda}
\frac
{\partial y^\gamma}
{\partial x^\mu}
\Biggl(
\Bigl\{
\begin{matrix}
\epsilon \\
\beta\gamma
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\epsilon}
\Biggr)
A_\epsilon
\end{split}    \\
\intertext
{\texttt
{... so ...
}
}
\begin{split}
\frac{\partial C_\lambda}{\partial x^\mu}
-
\Biggl(
\Bigl\{
\begin{matrix}
\nu \\
\lambda\mu
\end{matrix}
\Bigr\}
&
+
\frac{1}{2}
S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu}
\Biggr)
C_\nu
=    \\
&
+
\frac{\partial y^\epsilon}{\partial x^\lambda}
\frac{\partial y^\gamma}{\partial x^\mu}
\Biggl(
\frac{\partial A_\epsilon}{\partial y^\gamma}
-
\biggl(
\Bigl\{
\begin{matrix}
\alpha \\
\epsilon\gamma
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{\epsilon\gamma\,.}^{\,\,\,\,\,\,\,\alpha}
\biggr)
A_\alpha
\Biggr)
\end{split}    \\
\end{align*}
\end{gather*}



\begin{gather*}
\intertext
{\texttt
{...
\newline
...
\newline
... derive $B_{\,.\,mnp}^k$ ...
\newline
...
\newline
... covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ...
}
}
\begin{align*}
\begin{split}
A_{m,n}
=
&
+
\frac{\partial A_m}{\partial x^n}
-
\Gamma_{mn}^{\,\,k}
A_k
\end{split}    \\
\intertext
{\texttt
{... $2^{nd}$ covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ...
}
}
\begin{split}
A_{m,np}
=
&
+
\frac{\partial A_{m,n}}{\partial x^p}    \\
&
-
\Gamma_{mp}^{\,\,s}
A_{s,n}    \\
&
-
\Gamma_{np}^{\,\,s}
A_{m,s}    \\
=
&
+
\frac{\partial^2 A_m}{\partial x^n\partial x^p}
-
\frac
{
\partial
\Gamma_{mn}^{\,\,k}
}
{
\partial x^p
}
A_k
-
\Gamma_{mn}^{\,\,k}
\frac{\partial A_k}{\partial x^p}    \\
&
-
\Gamma_{mp}^{\,\,s}
\frac{\partial A_s}{\partial x^n}
+
\Gamma_{mp}^{\,\,s}
\Gamma_{sn}^{\,k}
A_k    \\
&
-
\Gamma_{np}^{\,\,s}
A_{m,s}
\end{split}    \\
\intertext
{\texttt
{... so (a tensor) ...
}
}
\begin{split}
A_{m,np}
-
A_{m,pn}
=
&
+
\frac{\partial^2 A_m}{\partial x^n\partial x^p}
-
\frac
{
\partial
\Gamma_{mn}^{\,\,k}
}
{
\partial x^p
}
A_k
-
\Gamma_{mn}^{\,\,k}
\frac{\partial A_k}{\partial x^p}    \\
&
-
\Gamma_{mp}^{\,\,s}
\frac{\partial A_s}{\partial x^n}
+
\Gamma_{mp}^{\,\,s}
\Gamma_{sn}^{\,k}
A_k    \\
&
-
\Gamma_{np}^{\,\,s}
A_{m,s}    \\
&
-
\frac{\partial^2 A_m}{\partial x^p\partial x^n}
+
\frac
{
\partial
\Gamma_{mp}^{\,\,k}
}
{
\partial x^n
}
A_k
+
\Gamma_{mp}^{\,\,k}
\frac{\partial A_k}{\partial x^n}    \\
&
+
\Gamma_{mn}^{\,\,s}
\frac{\partial A_s}{\partial x^p}
-
\Gamma_{mn}^{\,\,s}
\Gamma_{sp}^{\,k}
A_k    \\
&
+
\Gamma_{pn}^{\,\,s}
A_{m,s}    \\
=
&
+
B_{\,.\,mnp}^k
A_k
+
S_{pn\,.}^{\,\,\,\,\,\,\,\,s}\,
A_{m,s}
\end{split}    \\
\intertext
{\texttt
{... where ...
}
}
\begin{split}
B_{\,.\,mnp}^k
=
&
+
\frac
{
\partial
\Gamma_{mp}^{\,\,k}
}
{
\partial x^n
}
-
\frac
{
\partial
\Gamma_{mn}^{\,\,k}
}
{
\partial x^p
}
+
\Gamma_{mp}^{\,\,s}
\Gamma_{sn}^{\,k}
-
\Gamma_{mn}^{\,\,s}
\Gamma_{sp}^{\,k}
\end{split}
\end{align*}
\end{gather*}



\begin{gather*}
\intertext
{\texttt
{...
\newline
...
\newline
... derive $B_{\,.\,mnp}^k$ ...
\newline
...
\newline
... covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ...
}
}
\begin{align*}
\begin{split}
A_{m,n}
=
&
+
\frac{\partial A_m}{\partial x^n}
-
\Gamma_{mn}^{\,\,k}
A_k
\end{split}    \\
\intertext
{\texttt
{... $2^{nd}$ covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ...
}
}
\begin{split}
A_{m,np}
=
&
+
\frac{\partial A_{m,n}}{\partial x^p}    \\
&
-
\Gamma_{mp}^{\,\,s}
A_{s,n}    \\
&
-
\Gamma_{np}^{\,\,s}
A_{m,s}    \\
=
&
+
\frac{\partial^2 A_m}{\partial x^n\partial x^p}
-
\frac
{
\partial
\Gamma_{mn}^{\,\,k}
}
{
\partial x^p
}
A_k
-
\Gamma_{mn}^{\,\,k}
\frac{\partial A_k}{\partial x^p}    \\
&
-
\Gamma_{mp}^{\,\,s}
\frac{\partial A_s}{\partial x^n}
+
\Gamma_{mp}^{\,\,s}
\Gamma_{sn}^{\,k}
A_k    \\
&
-
\Gamma_{np}^{\,\,s}
A_{m,s}
\end{split}    \\
\intertext
{\texttt
{... so (a tensor) ...
}
}
\begin{split}
A_{m,np}
-
A_{m,pn}
=
&
+
\frac{\partial^2 A_m}{\partial x^n\partial x^p}
-
\frac
{
\partial
\Gamma_{mn}^{\,\,k}
}
{
\partial x^p
}
A_k
-
\Gamma_{mn}^{\,\,k}
\frac{\partial A_k}{\partial x^p}    \\
&
-
\Gamma_{mp}^{\,\,s}
\frac{\partial A_s}{\partial x^n}
+
\Gamma_{mp}^{\,\,s}
\Gamma_{sn}^{\,k}
A_k    \\
&
-
\Gamma_{np}^{\,\,s}
A_{m,s}    \\
&
-
\frac{\partial^2 A_m}{\partial x^p\partial x^n}
+
\frac
{
\partial
\Gamma_{mp}^{\,\,k}
}
{
\partial x^n
}
A_k
+
\Gamma_{mp}^{\,\,k}
\frac{\partial A_k}{\partial x^n}    \\
&
+
\Gamma_{mn}^{\,\,s}
\frac{\partial A_s}{\partial x^p}
-
\Gamma_{mn}^{\,\,s}
\Gamma_{sp}^{\,k}
A_k    \\
&
+
\Gamma_{pn}^{\,\,s}
A_{m,s}    \\
=
&
+
B_{\,.\,mnp}^k
A_k
+
S_{pn\,.}^{\,\,\,\,\,\,\,\,s}\,
A_{m,s}
\end{split}    \\
\intertext
{\texttt
{... where ...
}
}
\begin{split}
B_{\,.\,mnp}^k
=
&
+
\frac
{
\partial
\Gamma_{mp}^{\,\,k}
}
{
\partial x^n
}
-
\frac
{
\partial
\Gamma_{mn}^{\,\,k}
}
{
\partial x^p
}
+
\Gamma_{mp}^{\,\,s}
\Gamma_{sn}^{\,k}
-
\Gamma_{mn}^{\,\,s}
\Gamma_{sp}^{\,k}
\end{split}
\end{align*}
\end{gather*}



\begin{gather*}
\intertext
{\texttt
{...
\newline
...
\newline
...
\newline
... the metric tensor ...
\newline
...
\newline
... the covariant derivative $h_{ij,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ...
\newline
...
$
(
\frac
{\partial h_{ij}}
{\partial x^k}
-
\Gamma_{ik}^n
h_{nj}
-
\Gamma_{jk}^n
h_{in}
)
\neq
0
$
...
}
}
\begin{align*}
\begin{split}
C_{i,k}
=
&
+
\bigl(
h_{ij}
C^j
\bigr)_{,k}
\end{split}    \\
\begin{split}
\frac
{
\partial C_i
}
{
\partial x^k
}
-
\Gamma_{ik}^n
C_n
=
&
+
h_{ij}
\Bigl(
\frac
{
\partial C^j
}
{
\partial x^k
}
+
\Gamma_{nk}^j
C^n
\Bigr)    \\
&
+
\Bigl(
\frac
{\partial h_{ij}}
{\partial x^k}
-
\Gamma_{ik}^n
h_{nj}
-
\Gamma_{jk}^n
h_{in}
\Bigr)
C^j
\end{split}    \\
\begin{split}
\frac
{
\partial C_i
}
{
\partial x^k
}
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
ik
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
\Bigr)
C_n
=
&
+
h_{ij}
\Biggl(
\frac
{
\partial C^j
}
{
\partial x^k
}
+
\Bigl(
\Bigl\{
\begin{matrix}
j \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
\Bigr)
C^n
\Biggr)    \\
&
+
\Biggl(
\,
+
\frac
{\partial h_{ij}}
{\partial x^k}    \\
&
\qquad
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
ik
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
\Bigr)
h_{nj}    \\
&
\qquad
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
jk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{jk\,.}^{\,\,\,\,\,\,n}
\Bigr)
h_{in}    \\
&
\qquad
\Biggr)
C^j
\end{split}    \\
\intertext
{\texttt
{... so ...
}
}
\begin{split}
-
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
C_n
=
&
+
\frac{1}{2}
h_{ij}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
C^n    \\
&
-
\frac{1}{2}
h_{nj}
S_{ik\,.}^{\,\,\,\,\,\,n}
C^j    \\
&
-
\frac{1}{2}
h_{in}
S_{jk\,.}^{\,\,\,\,\,\,n}
C^j
\end{split}    \\
\begin{split}
-
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
C_n
=
&
-
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
C_n
\end{split}    \\
\intertext
{\texttt
{...
\newline
... the covariant derivative $h^{ij}_{\,\,\,\,,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ...
\newline
...
$
(
\frac
{\partial h^{ij}}
{\partial x^k}
+
\Gamma_{nk}^{\,\,i}
h^{nj}
+
\Gamma_{nk}^{\,\,j}
h^{in}
)
\neq
0
$
...
}
}
\begin{split}
C_{\,,k}^i
=
&
+
\bigl(
h^{ij}
C_j
\bigr)_{,k}
\end{split}    \\
\begin{split}
\frac
{
\partial C^i
}
{
\partial x^k
}
+
\Gamma_{nk}^{\,\,i}
C^n
=
&
+
h^{ij}
\Bigl(
\frac
{
\partial C_j
}
{
\partial x^k
}
-
\Gamma_{jk}^n
C_n
\Bigr)    \\
&
+
\Bigl(
\frac
{\partial h^{ij}}
{\partial x^k}
+
\Gamma_{nk}^{\,\,i}
h^{nj}
+
\Gamma_{nk}^{\,\,j}
h^{in}
\Bigr)
C_j
\end{split}    \\
\begin{split}
\frac
{
\partial C^i
}
{
\partial x^k
}
+
\Bigl(
\Bigl\{
\begin{matrix}
i \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
\Bigr)
C^n
=
&
+
h^{ij}
\Biggl(
\frac
{
\partial C_j
}
{
\partial x^k
}
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
jk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{jk\,.}^{\,\,\,\,\,\,n}
\Bigr)
C_n
\Biggr)    \\
&
+
\Biggl(
\,
+
\frac
{\partial h^{ij}}
{\partial x^k}    \\
&
\qquad
+
\Bigl(
\Bigl\{
\begin{matrix}
i \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
\Bigr)
h^{nj}    \\
&
\qquad
+
\Bigl(
\Bigl\{
\begin{matrix}
j \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
\Bigr)
h^{in}    \\
&
\qquad
\Biggr)
C_j
\end{split}    \\
\intertext
{\texttt
{... so ...
}
}
\begin{split}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C^n
=
&
-
\frac{1}{2}
h^{ij}
S_{jk\,.}^{\,\,\,\,\,\,n}
C_n    \\
&
+
\frac{1}{2}
h^{nj}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C_j    \\
&
+
\frac{1}{2}
h^{in}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
C_j
\end{split}    \\
\begin{split}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C^n
=
&
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C^n
\end{split}    \\
\end{align*}
\end{gather*}



\begin{gather*}
\intertext
{\texttt
{...
\newline
...
\newline
...
\newline
... the metric tensor ...
\newline
...
\newline
... the covariant derivative $h_{ij,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ...
\newline
...
$
(
\frac
{\partial h_{ij}}
{\partial x^k}
-
\Gamma_{ik}^n
h_{nj}
-
\Gamma_{jk}^n
h_{in}
)
\neq
0
$
...
}
}
\begin{align*}
\begin{split}
C_{i,k}
=
&
+
\bigl(
h_{ij}
C^j
\bigr)_{,k}
\end{split}    \\
\begin{split}
\frac
{
\partial C_i
}
{
\partial x^k
}
-
\Gamma_{ik}^n
C_n
=
&
+
h_{ij}
\Bigl(
\frac
{
\partial C^j
}
{
\partial x^k
}
+
\Gamma_{nk}^j
C^n
\Bigr)    \\
&
+
\Bigl(
\frac
{\partial h_{ij}}
{\partial x^k}
-
\Gamma_{ik}^n
h_{nj}
-
\Gamma_{jk}^n
h_{in}
\Bigr)
C^j
\end{split}    \\
\begin{split}
\frac
{
\partial C_i
}
{
\partial x^k
}
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
ik
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
\Bigr)
C_n
=
&
+
h_{ij}
\Biggl(
\frac
{
\partial C^j
}
{
\partial x^k
}
+
\Bigl(
\Bigl\{
\begin{matrix}
j \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
\Bigr)
C^n
\Biggr)    \\
&
+
\Biggl(
\,
+
\frac
{\partial h_{ij}}
{\partial x^k}    \\
&
\qquad
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
ik
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
\Bigr)
h_{nj}    \\
&
\qquad
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
jk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{jk\,.}^{\,\,\,\,\,\,n}
\Bigr)
h_{in}    \\
&
\qquad
\Biggr)
C^j
\end{split}    \\
\intertext
{\texttt
{... so ...
}
}
\begin{split}
-
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
C_n
=
&
+
\frac{1}{2}
h_{ij}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
C^n    \\
&
-
\frac{1}{2}
h_{nj}
S_{ik\,.}^{\,\,\,\,\,\,n}
C^j    \\
&
-
\frac{1}{2}
h_{in}
S_{jk\,.}^{\,\,\,\,\,\,n}
C^j
\end{split}    \\
\begin{split}
-
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
C_n
=
&
-
\frac{1}{2}
S_{ik\,.}^{\,\,\,\,\,\,n}
C_n
\end{split}    \\
\intertext
{\texttt
{...
\newline
... the covariant derivative $h^{ij}_{\,\,\,\,,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ...
\newline
...
$
(
\frac
{\partial h^{ij}}
{\partial x^k}
+
\Gamma_{nk}^{\,\,i}
h^{nj}
+
\Gamma_{nk}^{\,\,j}
h^{in}
)
\neq
0
$
...
}
}
\begin{split}
C_{\,,k}^i
=
&
+
\bigl(
h^{ij}
C_j
\bigr)_{,k}
\end{split}    \\
\begin{split}
\frac
{
\partial C^i
}
{
\partial x^k
}
+
\Gamma_{nk}^{\,\,i}
C^n
=
&
+
h^{ij}
\Bigl(
\frac
{
\partial C_j
}
{
\partial x^k
}
-
\Gamma_{jk}^n
C_n
\Bigr)    \\
&
+
\Bigl(
\frac
{\partial h^{ij}}
{\partial x^k}
+
\Gamma_{nk}^{\,\,i}
h^{nj}
+
\Gamma_{nk}^{\,\,j}
h^{in}
\Bigr)
C_j
\end{split}    \\
\begin{split}
\frac
{
\partial C^i
}
{
\partial x^k
}
+
\Bigl(
\Bigl\{
\begin{matrix}
i \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
\Bigr)
C^n
=
&
+
h^{ij}
\Biggl(
\frac
{
\partial C_j
}
{
\partial x^k
}
-
\Bigl(
\Bigl\{
\begin{matrix}
n \\
jk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{jk\,.}^{\,\,\,\,\,\,n}
\Bigr)
C_n
\Biggr)    \\
&
+
\Biggl(
\,
+
\frac
{\partial h^{ij}}
{\partial x^k}    \\
&
\qquad
+
\Bigl(
\Bigl\{
\begin{matrix}
i \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
\Bigr)
h^{nj}    \\
&
\qquad
+
\Bigl(
\Bigl\{
\begin{matrix}
j \\
nk
\end{matrix}
\Bigr\}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
\Bigr)
h^{in}    \\
&
\qquad
\Biggr)
C_j
\end{split}    \\
\intertext
{\texttt
{... so ...
}
}
\begin{split}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C^n
=
&
-
\frac{1}{2}
h^{ij}
S_{jk\,.}^{\,\,\,\,\,\,n}
C_n    \\
&
+
\frac{1}{2}
h^{nj}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C_j    \\
&
+
\frac{1}{2}
h^{in}
S_{nk\,.}^{\,\,\,\,\,\,\,\,j}
C_j
\end{split}    \\
\begin{split}
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C^n
=
&
+
\frac{1}{2}
S_{nk\,.}^{\,\,\,\,\,\,\,\,i}
C^n
\end{split}    \\
\end{align*}
\end{gather*}



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Contributed by TomAllen

Essays/Torsion Tensor/TorsionTensor02 (last edited 2010-09-01 04:40:37 by TomAllen)