Abstract. We study the fth term in the asymptotic expansionof the heat operator trace for a partial dierential operator of Laplace type on a compact Riemannian manifold with Dirichlet or Neumann boundary conditions. (n m)=2 a n (F;D;B) where the a n (F;D;B) are locally computable; see [6] for details. We computed a n for n 4 in [3]; we h a v e c hanged notation slightly from that paper. In this paper, we compute a 5 if the boundary is totally geodesic. If the boundary is not totally geodesic, the second fundamental form L enters and the combinatorial complexity becomes formidable. In this setting, we also obtain a result by restricting attention 1991 Mathematics Subject Classication. Primary 58G25.
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