A Note on the Square of Dirac Type Differential Operators
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Submission date: 24. Aug. 2006
published in: Journal of geometry and physics, 57 (2007) 10, p. 1999-2013
DOI number (of the published article): 10.1016/j.geomphys.2007.04.004
with the following different title: On the square of first order differential operators of Dirac type and the Einstein-Hilbert action
MSC-Numbers: 53C05, 53C07
PACS-Numbers: 02.40.Hw, 02.40.Ma
Keywords and phrases: Clifford Module Bundles, Dirac Type Differential Operators, Linear Connections
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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.