The stretch of a foliation and geometric superrigidity

Quiroga-Barranco, Raul

doi:10.1090/S0002-9947-97-01732-7

The stretch of a foliation and geometric superrigidity
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by Raul Quiroga-Barranco PDF

Trans. Amer. Math. Soc. 349 (1997), 2391-2426 Request permission

Abstract:

We consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type. Such objects can be considered as compact models for the geometry of the symmetric space. Based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions. To achieve this we consider the heat flow equation along the leaves of a foliation, a Bochner formula on foliations and a geometric invariant for foliations with leafwise Riemannian metrics called the stretch. We obtain as applications a metric rigidity theorem for foliations and a rigidity type result for Riemannian manifolds whose geometry is only partially symmetric.

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