An Explicit Kaluza-Klein Reduction of Einstein's Gravity in $6D$ on $S^2$

We study a six-dimensional Kaluza-Klein theory with spacetime topology $M_4 \times S^2$ and analyze the gauge sector arising from dimensional reduction. Using normalized Killing vectors on $S^2$, we explicitly construct the reduced Yang-Mills action and determine the corresponding gauge kinetic matrix. Despite the $SO(3)$ isometry of $S^2$, we show that only two physical gauge fields propagate in four dimensions. The gauge kinetic matrix therefore has rank two and possesses a single zero eigenvalue. We demonstrate that this degeneracy is a direct consequence of the coset structure $S^2 \simeq SO(3)/SO(2)$ and reflects a non-dynamical gauge direction rather than an inconsistency of the reduction. Our results clarify the geometric origin of gauge degrees of freedom in Kaluza-Klein reductions on coset spaces.