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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Nhikawa, S., Tondeur, P. & Vanhecke, L. Spectral geometry for Riemannian foliations. Ann Glob Anal Geom 10, 291–304 (1992). https://doi.org/10.1007/BF00136871
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DOI: https://doi.org/10.1007/BF00136871