The stretch of a foliation and geometric superrigidity. (English) Zbl 0878.53026
Author’s 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”.
Reviewer: A.P.Stone (Albuquerque)
MSC:
| 53C12 | Foliations (differential geometric aspects) |
| 58E20 | Harmonic maps, etc. |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 28A33 | Spaces of measures, convergence of measures |
Keywords:
symmetric space; superrigidity; totally geodesic immersions; heat flow; foliations; leafwise Riemannian metrics; rigidityReferences:
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