  We first exhibit in the commutative case the simple algebraic relations
between the algebra of functions on a manifold and its infinitesimal length
element $ds$. Its unitary representations correspond to Riemannian metrics and
Spin structure while $ds$ is the Dirac propagator $ds = \ts \!\!$---$\!\! \ts =
D^{-1}$ where $D$ is the Dirac operator. We extend these simple relations to
the non commutative case using Tomita's involution $J$. We then write a
spectral action, the trace of a function of the length element in Planck units,
which when applied to the non commutative geometry of the Standard Model will
be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian
coupled to gravity. The internal fluctuations of the non commutative geometry
are trivial in the commutative case but yield the full bosonic sector of SM
with all correct quantum numbers in the slightly non commutative case. The
group of local gauge transformations appears spontaneously as a normal subgroup
of the diffeomorphism group.
