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A Note on the Square of Dirac Type Differential Operators
The aim of this note is to present a new global formula for the Lichnerowicz decomposition of a general Dirac type first order differential operator. This formula generalizes the well-known Lichnerowicz formula for Dirac type operators which are determined by Clifford connections on an arbitrary Clifford module bundle.
Moreover, we also show that the connection class of a general Dirac type operator has a natural representative. In this sense, each Dirac type operator defines a natural connection on a given Clifford module bundle. This generalizes the fact that the connection class of a Dirac type operator possesses at most one Clifford connection.