Heat kernel expansions in vector bundles |
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Authors: | Martin N Ndumu |
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Institution: | Department of Mathematics and Computer Science, University of Maryland Eastern Shore, Princess Anne, MD 21853, United States |
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Abstract: | Let M be a complete connected smooth (compact) Riemannian manifold of dimension n. Let Π:V→M be a smooth vector bundle over M. Let be a second order differential operator on M, where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M. Let kt(−,−) be the heat kernel of V relative to L. In this paper we will derive an exact and an asymptotic expansion for kt(x,y0) where y0 is the center of normal coordinates defined on M, x is a point in the normal neighborhood centered at y0. The leading coefficients of the expansion are then computed at x=y0 in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M, of the vector bundle V, and of the vector bundle section ? and its derivatives.We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean-Singer). |
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Keywords: | 58J35 58J65 |
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