In this study, we investigate the underlying geometry of the gauge principle as an alternative way for understanding particle interactions. We employ the linear sigma model, based on the global–local symmetry group \(O(N+1)\) , reducing to O(N) after a procedure of spontaneous symmetry breaking. We illustrate how the dynamics of the Goldstone bosons—arising from the spontaneous symmetry breaking—are governed by the covariant derivative, which is derived from first principles using Killing vectors notion. Besides, we analyze how the internal space acquires a curvature when locality concept is demanded into the theory. The curvature tensor for the manifold of the internal space is precisely the field strength tensor, which is used to formulate the kinetic term of the bosons in gauge quantum field theories.
