A Remark on the Regularity of the }-function of Dirac Operators

Recently, Bismut[ 2] proved a new local index theorem for Dirac operators associated with certain non-torsion-free connections. In this paper, we discuss the odd dimensional analogue. Here the object is the 17-function introduced by Atiyah, Patodi and Singer[l]. And we prove the regularity property of the ~7-function of Dirac operators associated with the appropriate non-torsion-free connections. The paper is organized as follows: In w we review some notation and formulas in [2] and we reproved the Bismut local index theorem for D L +c B by using Getzler-Yu method ([5], [6]). The even and odd dimensional cases of Dirac operators with non-torsion-free connections will be discussed in w