The asymptotic expansion for the trace of the heat kernel on a generalized surface of revolution

Abstract

Let M M be a smooth compact Riemannian manifold without boundary. Let I I be an open interval. Let h ( r ) h(r) be a smooth positive function. Let g g be the metric on M M . Consider the fundamental solution E ( x , y , r 1 , r 2 ; t ) E(x,y,{r_1},{r_2};t) of the heat equation on M × I M \times I with metric h 2 ( r ) g + d r ⊗ d r {h^2}(r)g + dr \otimes dr (when E E exists globally we call it the heat kernel on M × I M \times I ). The coefficients of the asymptotic expansion of the trace E E are studied and expressed in terms of corresponding coefficients on the basis M M . It is fulfilled by means of constructing a parametrix for E E which is different from a parametrix in the standard form. One important result is that each of the former coefficients is a linear combination of the latter coefficients.