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

Abstract:

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