The heat kernel associated with an elliptic second-order partial differential
operator of Laplace type acting on smooth sections of a vector bundle over a
Riemannian manifold, is studied. A general manifestly covariant method for
computation of the coefficients of the heat kernel asymptotic expansion is
developed. The technique enables one to compute explicitly the diagonal values
of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De
Witt-Seeley coefficients, as well as their derivatives. The elaborated
technique is applicable for a manifold of arbitrary dimension and for a generic
Riemannian metric of arbitrary signature. It is very algorithmic, and well
suited to automated computation. The fourth heat kernel coefficient is computed
explicitly for the first time. The general structure of the heat kernel
coefficients is investigated in detail. On the one hand, the leading derivative
terms in all heat kernel coefficients are computed. On the other hand, the
generating functions in closed covariant form for the covariantly constant
terms and some low-derivative terms in the heat kernel coefficients are
constructed by means of purely algebraic methods. This gives, in particular,
the whole sequence of heat kernel coefficients for an arbitrary locally
symmetric space.Comment: 31 pages, LaTeX, no figures, Invited Lecture at the University of
Iowa, Iowa City, April, 199