Warped Geometry in Higher Dimensions with an Orbifold Extra Dimension

We solve the Einstein equations in higher dimensions with warped geometry where an extra dimension is assumed to have orbifold symmetry $S^{1}/Z_{2}$. The setup considered here is an extension of the five-dimensional Randall-Sundrum model to $5+D$ dimensions, and hidden and observable branes are fixed on the orbifold. It is assumed that the brane tension (self-energy) of each brane with $(4+D)$-dimensional spacetime is anisotropic and that the warped metric function of the four dimensions is generally different from that of the extra $D$ dimensions. We point out that the forms of the warped metric functions and the relations between the tensions of two branes depend on the integration constant appearing in the Einstein equations as well as on the sign of the bulk cosmological constant.