1,118 research outputs found

    Traces of heat operators on Riemannian foliations

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    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.Comment: 37 pages, citations update

    Riemannian flows and adiabatic limits

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    We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow

    A brief note on the spectrum of the basic Dirac operator

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    In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation (M,F)(M,\mathcal{F}) with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac operator on MM. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric