with the assumption of direct product structure of the spacetime, one can get reduced action from the total action (\[1eq1\]). This reduced action is an $m$-dimensional version of Lovelock gravity as given below in Eq. (\[reducedaction\]). We can get equations of motion (\[emm\]) only for $m$-dimensional part but not the $n$-dimensional part by variation of this reduced action. Obviously these are nothing but the usual equations of motion of the Lovelock gravity in $m$ dimensions with special coefficients $A_{t}$’s. For example, when $m=4$, these equations are the Einstein equations with cosmological constant $$\label{eins}
A_0 g_{ij} + A_1 E_{ij}=0\, ,$$ where $E_{ij}$ is the Einstein tensor in four dimensions, and $A_0$ and $A_1$ are given in (\[lineq\]). Because our solutions are obtained for $A_0=A_1=0$, Eqs. (\[eins\]) have no information on our solutions. Therefore the field equations for $m$-dimensional part is trivially satisfied, while the nontrivial solutions come from the trace equation of gravitational field for the $n$-dimensional part, which is not obtained from the reduced action. So the reduced action certainly exists, but it does not give solutions in this paper. If we naively omit the extra dimensions in our solutions, the corresponding $m$-dimensional local diffeomorphism-invariant “effective action" is absent, and 