An explicit Kaluza–Klein reduction of einstein’s gravity in 6D on S2

Abstract

We study a six-dimensional Kaluza–Klein theory with spacetime topology M4 × S2 and analyze the gauge sector arising from dimensional reduction. Using normalized Killing vectors on S2, we explicitly construct the reduced Yang–Mills action and determine the corresponding gauge kinetic matrix. Despite the SO(3) isometry of S2, 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 S2 ≃ 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.