Spectral geometry for Riemannian foliations |
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Authors: | Seiki Nhikawa Philippe Tondeur Lieven Vanhecke |
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Institution: | (1) Mathematical Institute, Tôhoku University, 980 Sendai, Japan;(2) Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, 61801 Urbana, Illinois, U.S.A.;(3) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium |
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Abstract: | Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants? |
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Keywords: | Riemannian foliation curvature Jacobi operator spectral invariants |
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