Symmetry of Generalized Randall-Sundrum Model and Distribution of 3-Branes in Six-Dimensional Spacetime

A generalization from the usual $5$-dimensional two-brane Randall-Sundrum (RS) model to a $6$-dimensional multi-brane RS model is presented. The extra dimensions are extended from one to two; correspondingly the single-variable warp function is generalized to be a double-variable function, to represent the two extra dimensions. In the analysis of the Einstein equation we have two remarkable discoveries. One is that, when branes are absent, the cosmological parameter distributed in the two extra dimensions acts as a function describing a family of circles. These circles are not artificially added ones but stem from the equations of motion, while their radii are inversely proportional to the square root of the cosmological parameter. The other discovery is that, on any circle, there symmetrically distribute four branes. Their tensions, $V_1 sim V_4$, satisfy a particular relationship $V_1=V_3=-V_2=-V_4=3M^4$, where $M$ is the $6$-dimensional fundamental scale of the RS model.