Abstract

We prove a finiteness theorem for the spectral sequence (Ei(∇), (d)i) associated to a Riemannian foliation [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] on a compact manifold M, and to a flat vector bundle E over M with flat connection ∇. Using this result we prove that every Riemannian foliation on a compact manifold is tense (in the sense of [F. W. Kamber and Ph. Tondeur, Foliations and metrics, Progr. Math., vol. 32, 1983, pp. 103-152]). We also show that the main tautness theorems for Riemannian foliations on compact manifolds, which were proved by several authors, are immediate consequences of our results.

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