In this paper the authors first give an alternative short proof of the local index theorem by combining the pseudodifferential Greiner's representation of the heat kernel (Gr) with the Getzler rescalling. Extending the arguments of the proof the authors show that the Connes-Moscovici cocycle, representing the cyclic cohomology Chern character, for Diract operators on a compact spin manifold is exactly given by the A-form of the Riemann curvature. In the even dimensional case the authors recover the local form of the Atiyah-Singer index theorem and in the odd dimensional case the authors get back the spectral flow theorem of Atiyah-Patodi-Singler (APS).
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