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Structural aspects of the exponential expansion of the heat kernel. (English) Zbl 0866.58056
Fulling, S. A. (ed.), Heat kernel technique and quantum gravity. Proceedings of a conference, Winnipeg, Canada, August 2–6 1994. College Station, TX: Texas A & M University. Discourses Math. Appl. 4, 101-114 (1995).
Summary: (1) The small-time expansion of the logarithm of the heat kernel of a scalar operator contains only terms that are “connected” by tensor contractions. We offer a simple proof of this fact based on the necessary structure of a heat kernel on a product manifold. This result generalizes and systematizes previous work, including the connectedness property of the graph expansions of Osborn and coworkers and the observation of Parker and coworkers about the role of the curvature scalar. (2) For a matrix-valued (vector bundle) operator the expansion does not exhibit such an obvious connectedness property. However, we show, using a theorem of Friedrichs in the representation theory of Lie algebras, that an analogous property does persist in the bundle case when tensor contractions are augmented by Lie products. (3) It follows that the terms of the small-time expansion of the heat kernel itself may be decomposed into a direct sum of connected and disconnected components, and that explicit information on the highly disconnected components at any order is provided by only the low-order terms in the expansion of the logarithm.
For the entire collection see [Zbl 0845.00044].
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J37 Perturbations of PDEs on manifolds; asymptotics