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TRACES OF HEAT OPERATORS ON RIEMANNIAN FOLIATIONS

KEN RICHARDSON
Transactions of the American Mathematical Society
Vol. 362, No. 5 (MAY 2010), pp. 2301-2337
Stable URL: http://www.jstor.org/stable/25677786
Page Count: 37
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TRACES OF HEAT OPERATORS ON RIEMANNIAN FOLIATIONS
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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 → 0. The coefficients of t α and of t α (log t) β 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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