A unitary invariant in Riemannian geometry

We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo--Kobayashi--Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.