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An index theorem for foliations. (English) Zbl 0592.58049
One of the fundamental problems in foliation theory is to relate the transverse geometry to its topological invariants: the exotic classes. Exotic characteristic classes of a foliation are cohomology classes associated to the cohomology of the classifying space of the foliation. In this paper the author proves a theorem relating the index theory to the exotic characteristic classes, in the case of a Riemannian foliation generated by \({\mathbb{R}}^ n\) acting locally and freely by isometries with constant orbit dimension on a compact oriented Riemannian manifold. He considers an elliptic differential operator P transversally invariant to this action, he constructs, using index (P), an \({\mathbb{R}}/{\mathbb{Z}}\) analytic invariant called virtindex (P) and he relates it to the Simons characteristic numbers of the Riemannian foliation. Applications of this result are given in studying the problem of an extension from an \({\mathbb{R}}\)-action to an \({\mathbb{R}}^ n\) action \((n>2)\).
Reviewer: I.Cattaneo

58J22 Exotic index theories on manifolds
58A30 Vector distributions (subbundles of the tangent bundles)