A black hole solution of higher-dimensional Weyl-Yang-Kaluza-Klein theory

We consider the Weyl$-$Yang gauge theory of gravitation in a $(4+3)$-dimensional curved space-time within the scenario of the non-Abelian Kaluza$-$Klein theory for the source and torsion-free limits. The explicit forms of the field equations containing a new spin current term and the energy-momentum tensors in the usual four dimensions are obtained through the well-known dimensional reduction procedure. In this limit, these field equations admit (anti-)dyon and magnetic (anti-)monopole solutions as well as non-Einsteinian solutions in the presence of a generalized Wu$-$Yang ansatz and some specific warping functions when the extra dimensions associated with the round and squashed three-sphere $S^3$ are, respectively, included. The (anti-)dyonic solution has similar properties to those of the Reissner$-$Nordstr"{o}m$-$de Sitter black holes of the Einstein$-$Yang$-$Mills system. However, the cosmological constant naturally appears in this approach, and it associates with the constant warping function as well as the three-sphere radius. It is demonstrated that not only the squashing parameter $\mathrm{\ell }$ behaves as the constant charge but also its sign can determine whether the solution is a dyon/monopole or an antidyon/antimonopole. It is also shown by using the power series method that the existence of nonconstant warping function is essential for finding new exact Schwarzschild-like solutions in the considered model.