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-rôle in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that thhis gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory.
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