can be reduced to simple master equations if we decompose the metric perturbation $\delta g_{\mu\nu}=h_{\mu\nu}$ into tensor-type, vector-type and scalar-type components, according to their transformation behaviour as tensors on $\K^n$. Now, we present these equations for the case of stationary perturbations.

Tensor perturbations
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It is convenient to expand tensor perturbations in terms of the eigentensors satisfying where $\hat \triangle_L$ is the Lichnerowicz operator Here, $\hat D_i$ and $\hat R_{ijkl}$ represent the covariant derivative and the curvature tensor with respect to the metric $\gamma_{ij}$ on $\K^n$, respectively. When $\K^n$ is a constant curvature space, $\hat\triangle_L$ can be written simply as Hence, $k_T^2$ corresponds to the eigenvalue of the operator $-\hat 
D\cdot\hat D$ on second-rank symmetric tensors. In this case, $k_T^2$ takes discrete and positive values, except in the case $K=0$. In particular, for $\K^n=S^n$, its spectrum is given by In contrast, for $K=0$, i.e., when $\K^n$ is a torus $T^n$, the smallest eigenvalue is zero, and the corresponding eigentensors are given by trace-free constant symmetric matrices. There exist $n(n+1)/2-1$ such matrices, and these matrices represent variations of the moduli parameters of $T^n$.

In general, the moduli degrees of freedom of a constant curvature space correspond to the tensor harmonics satisfying 