We prove a generalized version of the well-known Lichnerowicz formula for the
square of the most general Dirac operator D\ on an
even-dimensional spin manifold associated to a metric connection
∇. We use this formula to compute the subleading term
Φ1(x,x,D2)\ of the heat-kernel expansion of
D2. 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.Comment: 10 pages, plain te