  The well-known Yang-Mills theory with one $ S^{1} / Z_{2}$ universal extra
dimension (UED) is generalized to an arbitrary number of spatial extra
dimensions through a novel compactification scheme. In this paper, the
Riemannian flat based manifold under consideration contains $ n $ spatial extra
dimensions defined by $ n $ copies of the orbifold $ S^{1} / Z_{2}$. In this
approach, we present the gauge structure and the mass spectrum of the effective
four-dimensional theory. We introduce the concept of standard and nonstandard
gauge transformations of the effective theory, and explicitly identify the
emergence of massive vector fields in the same number as massless
('pseudo-Goldstone') scalars in the compactified theory, verifying that a
Higgs-like mechanism operates in the compactification process. It is found
that, in contrast with the one UED scenario, in cases with two or more UEDs
there emerge massive scalar fields. Besides, at phase-space level, the
Hamiltonian analysis yields that the higher-dimensional and compactified
theories are classically equivalent using the fundamental concept of canonical
transformation. This work lays the ground for a wider study on these theories
concerning their quantization and predictive power at the level of quantum
fluctuations.
