Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry

We compute the metric associated to 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 fibres is investigated. When one of the triple describes a manifold, one find 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 metric component is given by the internal part of the geometry. In the standard model, this extra component comes from the Higgs field.