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 (D) over tilde on an even-dimensional spin manifold associated to a metric connection <(de l)over tilde>. We use this formula to compute the subleading term Phi( 1)(x, x, (D) over tilde(2)) of the heat-kernel expansion of (D) over t ilde(2). The trace of this term plays a key role in the definition of a (euclidian) gravity action in the context of non-commutative geometr y. We show that this gravity action can be interpreted as defining a m odified euclidian Einstein-Cartan theory.