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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The stretch of a foliation and
geometric superrigidity


Author: Raul Quiroga-Barranco
Journal: Trans. Amer. Math. Soc. 349 (1997), 2391-2426
MSC (1991): Primary 53C12, 58E20; Secondary 58G11, 28A33
MathSciNet review: 1373646
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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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Additional Information

Raul Quiroga-Barranco
Affiliation: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, CIEA-IPN, Apartado Postal 14-740, 07300 Mexico DF, Mexico
Email: quiroga@math.cinvestav.mx

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01732-7
PII: S 0002-9947(97)01732-7
Received by editor(s): December 15, 1994
Received by editor(s) in revised form: December 2, 1995
Additional Notes: Research supported by the Andrew Corporation, CONACYT-Mexico, COFAA-IPN-Mexico and SNI-Mexico.
Article copyright: © Copyright 1997 American Mathematical Society