Here we prove the identity
\begin{equation}\label{eq:div} \begin{aligned} u_{i_1 \cdots i_k p}{}^{p} &= (\Delta u)_{i_1 \cdots i_k} + \sum_{s=1}^{k} \Bigl( R_{i_s p}{}^{pm} \, u_{i_1 \cdots i_{s-1} m\, i_{s+1} \cdots i_k} + \\ &\quad +2 \sum_{r < s} R_{i_s p\, i_r}{}^m \, u_{i_1 \cdots i_{r-1} m\, i_{r+1} \cdots i_{s-1}}{}^p{}_{i_{s+1} \cdots i_k}\Bigr), \end{aligned} \end{equation}mentioned in the proof of (0x67dbeaae) .
By the Ricci identity , it follows that
\[ u_{i_1 \cdots i_k p}{}^{p} = g^{pq} u_{i_1 \cdots i_k p\, ; q} = g^{pq} \bigl[ u_{i_1 \cdots p i_k \, ; q} +(R_{i_k p i_1}{}^{m} \, u_{m i_2 \cdots i_{k-1}})_{;q} + \cdots +(R_{i_k p i_{k-1}}{}^{m} u_{i_1 \cdots m})_{;q} \bigr], \]where the semicolon denotes differentiation (see (0x68c42f6b) ).
Since the curvature tensor of the sphere is covariantly constant, we deduce
\[ \begin{aligned} u_{i_1 \cdots i_k p}{}^{p} &= g^{pq} \bigl[ u_{i_1 \cdots p i_k \, ; q} +R_{i_k p i_1}{}^{m} \, u_{m i_2 \cdots i_{k-1}q} + \cdots +R_{i_k p i_{k-1}}{}^{m} u_{i_1 \cdots mq} \bigr]\\ &= u_{i_1 \cdots p i_k }{}^p +R_{i_k p i_1}{}^{m} \, u_{m i_2 \cdots i_{k-1}}{}^p + \cdots +R_{i_k p i_{k-1}}{}^{m} u_{i_1 \cdots m}{}^p \end{aligned} \]Employing Ricci’s identity once again yields
\[ u_{i_1 \cdots p i_k}{}^p = u_{i_1 \cdots p }{}^p{}_{i_k} +R_{i_k p i_1}{}^{m} \, u_{m i_2 \cdots i_{k-1}}{}^p + \cdots +R_{i_k p i_{k-1}}{}^{m} u_{i_1 \cdots m}{}^p +R_{i_k p}{}^{pm} u_{i_1\cdots i_{k-1} m} . \]Thus,
\[ u_{i_1 \cdots i_k p}{}^{p} = u_{i_1 \cdots p}{}^p{}_{i_k} +R_{i_k p}{}^{pm} u_{i_1\cdots i_{k-1} m} +2R_{i_k p i_1}{}^{m} \, u_{m i_2 \cdots i_{k-1}}{}^p + \cdots +2R_{i_k p i_{k-1}}{}^{m} u_{i_1 \cdots m}{}^p \]Since the curvature tensor is covariantly constant, we obtain \eqref{eq:div} by applying the same argument several times.