GRAVITY, NONCOMMUTATIVE GEOMETRY AND THE WODZICKI RESIDUE

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n greater than or equal to 4, n even, the Wodzicki residue Res(D--n+2) is the integral of the second coefficient of the heat kernel expansion of D-2, We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tenser product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.