A generalized Lichnerowicz formula, the Wodzicki Residue and Gravity

We prove a generalized version of the well-known Lichnerowicz formula for the square of the most general Dirac operator $\widetilde{D}$\ on an even-dimensional spin manifold associated to a metric connection $\widetilde{\nabla}$. We use this formula to compute the subleading term $\Phi_1(x,x, \widetilde{D}^2)$\ of the heat-kernel expansion of $\widetilde{D}^2$. The trace of this term plays a key-r$\hat {\petit\rm o}$le in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that this gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory.