On the heat equation and the index theorem

Abstract

The main error occurs on page 306 where it is implicitly assumed that the coefficients of the two operators A*A and AA* (associated to the signature operator A) are polynomial functions in the gij, their derivatives and (det g)-l . As we shall show later this is not quite t r u e t h e coefficients also involve d ] f ~ and the inverses of the principal minors of the matrix gu" Thus the form m in (5.1) is not a regular invariant of the metric in the sense of w 2, and so the Gilkey Theorem as formulated on p. 284 does not apply. To correct this we shall widen the notion of regularity (so as to include, in particular, the form ~o above) and then check that our proof of Gitkey's Theorem still holds in this wider context. In w regularity was only defined for invariants of a Riemann structure g (i.e. satisfying the naturality or invariance property (2.3)). It will perhaps make for greater clarity if we introduce our new notion of regularity for any function of g, independently of the invariance property. We shall say that f(g) is a regular function of g if, in any coordinate system, we have