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In Einstein-Gauss-Bonnet gravity, we study the quasi-normal modes (QNMs) ofthe tensor perturbation for the so-called Maeda-Dadhich black hole whichlocally has a topology $\mathcal{M}^n \simeq M^4 \times \mathcal{K}^{n-4}$. Ourdiscussion is based on the tensor perturbation equation derivedin~\cite{Cao:2021sty}, where the Kodama-Ishibashi gauge invariant formalism forEinstein gravity theory has been generalized to the Einstein-Gauss-Bonnetgravity theory. With the help of characteristic tensors for the constantcurvature space $\mathcal{K}^{n-4}$, we investigate the effect of extradimensions and obtain the scalar equation in four dimensional spacetime, whichis quite different from the Klein-Gordon equation. Using the asymptoticiteration method and the numerical integration method with the Kumaresan-Tuftsfrequency extraction method, we numerically calculate the QNM frequencies. Inour setups, characteristic frequencies depend on six distinct factors. They arethe spacetime dimension $n$, the Gauss-Bonnet coupling constant $\alpha$, theblack hole mass parameter $\mu$, the black hole charge parameter $q$, and two``quantum numbers" $l$, $\gamma$. Without loss of generality, the impact ofeach parameter on the characteristic frequencies is investigated while fixingother five parameters. Interestingly, the dimension of compactification parthas no significant impact on the lifetime of QNMs.

The perturbation in Einstein-Gauss-Bonnet gravity II: the quasi-normal modes of the tensor-type of the Kaluza-Klein black hole