Traces of heat operators on Riemannian foliations

Details Traces of heat operators on Riemannian foliations Traces of heat operators on Riemannian foliations

2007-10-05

arxiv.org/abs/0710.1324v2

Trans.Am.Math.Soc.362:2301-2337,2010

Mathematics

Differential Geometry

37 pages, citations updated

Scientific paper

10.1090/S0002-9947-09-05069-7

We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace of this operator has a particular short time asymptotic expansion. The coefficients 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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