ABSTRACT
The cancellation between the expansions at t = 0 of the traces of the two heat kernels in (In. 18) will be discussed using a rescaling argument due to Getzler. Although, as in [34], this can be carried out in local coordinates it is interpreted globally here. To do so the process of scaling, or rescaling, a vector bundle at a boundary hypersurface of a compact manifold with corners is examined. Such rescaling takes into account some jet information at the hypersurface. Getzler’s rescaling of the homomorphism bundle of the spinor bundle, lifted to the heat space, is of this type and will be used to prove the local index theorem. As usual the point of giving a general treatment (rather than just a computation in local coordinates) is that it can be carried over to other settings. In particular the extension to the 6setting is immediate. The integral defining the eta invariant is interpreted as a 6-integral in this way (in fact it converges absolutely). These rescaling arguments are used in the next chapter to discuss the convergence of the integral for the eta invariant on an odd dimensional exact 6-spin manifold and also to extend the analytic torsion of Ray and Singer (see [30]) to this class of complete Riemann manifolds. 8.1. Simple rescaling.
