We compute the metric associated with noncommutative spaces described by a tensor product of spectral triples. Well-known results of the two-sheets model (distance on a sheet, distance between the sheets) are extended to any product of two spectral triples. The distance between different points on different fibers is investigated. When one of the triples describes a manifold, one finds a Pythagorean theorem as soon as the direct sum of the internal states (viewed as projections) commutes with the internal Dirac operator. Scalar fluctuations yield a discrete Kaluza–Klein model in which the extra component of the metric is given by the internal part of the geometry. In the standard model, this extra component comes from the Higgs field.

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