$k_T^2=-2K$. Hence, these constant matrices exhaust all moduli degrees of freedom for $K=0$, and there exists no moduli degree of freedom for $K=1$. Further, from the integral identity we obtain for $K=-1$. Hence, for $K=-1$, there exist moduli degrees of freedom only when $n=2$, in accordance with Mostow’s rigidity theorem. Note that these tensor harmonics satisfying $k_T^2=-2K$ are the only non-trivial tensor harmonics for $n=2$.

Through the expansion in terms of these tensor harmonics, each mode of a stationary tensor perturbation can be expressed in terms of a gauge-invariant variable $H_T(r)$ as Note that stationary tensor perturbations are always static.

The linearised Einstein equations are reduced to where the prime denotes differentiation with respect to $r$, and Note that for the modes satisfying $k_T^2=-2K$, $H_T=\const$ is always a solution of this equation and represents a perturbation corresponding to the variation of the background metric with respect to its moduli parameters.

Vector perturbations {#subsec:BasicEqs:Vector}
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Stationary vector perturbations can be expanded in terms of the vector harmonics satisfying as where The eigenvalue $k_V^2$ takes discrete and positive values, except when $K=0$, and $k_V^2\ge n-1$ for $K=1$. In particular, for $\K^n=S^n$, the spectrum of $k_V^2$ is given by In contrast, when $K=0$ 