The Lichnerowicz formula and residue for Dirac operator with one form

In this paper, a Dirac operator with formal perturbation is studied. We study the decomposition formula of the Dirac operator squared, and get the generalized Lichnerowicz formula. According to the relation between the noncommutative remainder of a formal Dirac operator and the trace of the Dirac operator, the noncommutative remainder of a formal Dirac operator can be obtained by directly calculating the trace of the endomorphism.