Motion
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Consider a $D(=m+n)$-dimensional spacetime $X$, locally homeomorphic to $\mathcal{M}^m\times \mathcal{N}^n$. We assume the metric of this spacetime has the form $$\label{metric} ds^2=g_{ij}dx^i  dx^j+r_{0}^2 \gamma_{ab}dy^a dy^b\,
,$$ where $g_{ij}dx^i  dx^j$ is the metric on $\mathcal{M}^m$ with the coordinates $\{x^i,i=1,\cdots , m\}$, $r_{0}$ is a constant and $\gamma_{ab}dy^a  dy^b$ is the metric on the $n$-dimensional space $\mathcal{N}^n$ with the coordinates $\{y^a,a=,1,\cdots,n\}$. $\mathcal{N}^n$ is a constant curvature space with curvature $\bar{k}=\pm 1,0$. It is easy to find that the components of the Riemann tensor for (\[metric\]) have the form $$\begin{aligned}
\label{riemann1}
R_{ijkl}=\bar{R}_{ijkl},\quad R_{ij}{}{}^{kl}=\bar{R}_{ij}{}{}^{kl}\, ,\nonumber \\
R_{abcd}=\tilde{R}_{abcd},\quad
R_{ab}{}{}^{cd}=\tilde{R}_{ab}{}{}^{cd}\, ,\end{aligned}$$ where $\bar{R}_{ijkl}$ and $\tilde{R}_{abcd}$ denote the components of Riemann tensor on $\mathcal{M}^m$ and $\mathcal{N}^n$ respectively. For $\mathcal{N}^n$, we can write the $\widetilde{R}_{abcd}$ as $$\tilde{R}_{abcd}=\bar{k} r_{0}^2
\left(\gamma_{ac}\gamma_{bd}-\gamma_{ad}\gamma_{bc}\right), \quad
\tilde{R}^{ab}{}{}_{cd}= \frac{\bar{k}}{r_{0}^2} \delta^{ab}_{cd}\,
. \label{riemann2}$$ According to the decomposition of the Riemann tensor (\[riemann1\]), we can decompose the equations of motion (\[em\]) into $m$-dimensional and $n$-dimensional parts: $$\begin{aligned}
\mathcal{G}^{i}{}_{j}&\equiv&\sum_{k=0}^{p}\frac{1}{2^{k+1}}c_{k}\delta^{i
\mu_1\cdots \mu_k\nu_1\cdots\nu_k}_{j\lambda_1\cdots
\lambda_k\sigma_1\cdots\sigma_k}R^{\lambda_1\sigma_1}_{~~~~
\mu_1\nu_1}\cdots R^{\lambda_k\sigma_k}_{~~~~ \mu_k\nu_k}=0 \, , \nonumber \\
\mathcal{G}^{a}{}_{b}&\equiv&\sum_{k=0}^{p}\frac{1}{2^{k+1}}c_{k}\delta^{a
\mu_1\cdots \mu_k\nu_1\cdots\nu_k}_{b\lambda_1\cdots
\lambda_k\sigma_1\cdots\sigma_k}R^{\lambda_1\sigma_1}_{~~~~
\mu_1\nu_1}\cdots R^{\lambda_k\sigma_k}_{~~~~ \mu_k\nu_k}=0 \, .
\label{emsplit}\end{aligned}$$ Other components (such as $\mathcal{G}^{i}{}_{a}$) automatically vanish. Since $i,j$ run only in the range $\{1,\cdots ,
m\}$, and $R_{ijkl}$ can appear in the products “$RR\cdots$" no more 