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Motivated by Kaluza-Klein theory and modern string theories, the class of exact solutions yielding product manifolds M2 × S2 in general relativity is investigated. The compact submanifold homeomorphic to S2 is chosen to be a very small sphere. Choosing an anisotropic fluid as the particular physical model, it is proved that very large mass density and tension provide the mechanism of compactification. In case the transverse pressure is chosen to be zero, the corresponding spacetime is homeomorphic to ℝ2 × S2, and thus provides a tractable non-flat metric. In this simple metric, the geodesic equations are completely solved, yielding motions of massive test particles. Next, the corresponding wave mechanics (given by the Klein-Gordon equation) is explored in the same curved background. A general class of exact solutions is obtained. Four conserved quantities are explicitly computed. The scalar particles exhibit a discrete mass spectrum.