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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Traces of heat operators on Riemannian foliations
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by Ken Richardson PDF
Trans. Amer. Math. Soc. 362 (2010), 2301-2337 Request permission

Abstract:

We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace $K_{B}(t)$ of this operator has a particular asymptotic expansion as $t\to 0$. The coefficients of $t^{\alpha }$ and of $t^{\alpha }(\log t)^{\beta }$ in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl’s asymptotic formula. Also, we explicitly calculate the first two nontrivial coefficients of the expansion for special cases such as codimension two foliations and foliations with regular closure.
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Additional Information
  • Ken Richardson
  • Affiliation: Department of Mathematics, Texas Christian University, TCU Box 298900, Fort Worth, Texas 76129
  • Email: k.richardson@tcu.edu
  • Received by editor(s): October 8, 2007
  • Published electronically: December 8, 2009
  • Additional Notes: The author’s research at MSRI was supported in part by NSF grant DMS-9701755.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2301-2337
  • MSC (2010): Primary 53C12, 58J37, 58J35, 58J50
  • DOI: https://doi.org/10.1090/S0002-9947-09-05069-7
  • MathSciNet review: 2584602