  A classical result in differential geometry due to Lichnerowicz [8] is
concerned with the decomposition of the square of Dirac operators defined by
Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian
manifold $M$. Recently, this formula has been generalized to arbitrary Dirac
operators [2]. In this paper we prove a supersymmetric version of the
generalized Lichnerowicz formula, motivated by the fact that there is a
one-to-one correspondence between Clifford superconnections and Dirac
operators. We extend this result to obtain a simple formula for the
supercurvature of a generalized Bismut superconnection. This might be seen as a
first step to prove the local index theorem also for families of arbitrary
Dirac operators.
