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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 | References | Similar Articles | Additional Information

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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  • 1. W. Ballmann and P. Eberlein, Fundamental groups of manifolds of nonpositive curvature, J. Diff. Geom. 25 (1987), 1-22. MR 88b:53047
  • 2. W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Birkhäuser, Boston, 1985. MR 87h:53050
  • 3. K. Corlette and R. J. Zimmer, Superrigidity for cocycles and hyperbolic geometry, Internat. J. Math. 5 (1994), 273-290. MR 95g:58055
  • 4. C. Croke and A. Fathi, An inequality between energy and intersection, Bull. London Math. Soc. 22 (1990), 489-494. MR 92d:58042
  • 5. P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II, Acta Math. 149 (1982), 41-69. MR 83m:53055
  • 6. -, Symmetry diffeomorphism group of a manifold of nonpositive curvature, Trans. Amer. Math. Soc. 309 (1988), 355-374. MR 89i:53028
  • 7. P. Eberlein and J. Heber, A differential geometric characterization of symmetric spaces of higher rank, Inst. Hautes Études Sci. Publ. Math. 71 (1990), 33-44. MR 91j:53022
  • 8. S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Springer-Verlag, Berlin, 1987. MR 88k:53001
  • 9. C. Godbillon, Feuilletages, Études géométriques, Progress in Mathematics, vol. 98, Birkhäuser, Basel, 1991. MR 93i:57038
  • 10. M. Gromov, Rigid transformations groups, Géométrie Différentielle, Colloque Géométrie et Physique de 1986 en l'honneur de André Lichnerowicz (D. Bernard and Y. Choquet-Bruhat, eds.), Hermann, Paris, 1988, pp. 65-139. MR 90d:58173
  • 11. -, Foliated Plateau problem. Harmonic maps of foliations, Geom. Funct. Anal. 1 (1991), 253-320. MR 93a:58048
  • 12. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, New York, 1978. MR 80k:53081
  • 13. M. W. Hirsch, Differential topology, Springer-Verlag, New York, 1988. MR 56:6669 (1st ed.)
  • 14. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. I, Wiley, New York, 1963. MR 27:2945
  • 15. P. Li and L. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1-46. MR 93e:58039
  • 16. N. Mok, Y.-T. Siu, and S.-K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993), 57-83. MR 94h:53079
  • 17. C. C. Moore and C. Schochet, Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, vol. 9, Springer-Verlag, New York, 1988. MR 89h:58184
  • 18. G. D. Mostow, Strong rigidity for locally symmetric spaces, Annals of Mathematics Studies, vol. 78, Princeton University Press, Princeton, 1973. MR 52:5874
  • 19. A. L. Onishchik and E. B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin, 1990. MR 91g:22001
  • 20. K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, vol. 3, Academic Press, New York, 1967. MR 37:2271
  • 21. R. J. Spatzier and R. J. Zimmer, Fundamental groups of negatively curved manifolds and actions of semisimple groups, Topology 30 (1991), 591-601. MR 92m:57047
  • 22. R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. 112 (1980), 511-529. MR 82j:22011
  • 23. -, Ergodic theory, semisimple Lie groups and foliations by manifolds of negative curvature, Inst. Hautes Études Sci. Publ. Math. 55 (1982), 37-62. MR 84h:22022
  • 24. -, Ergodic theory and semisimple groups, Birkäuser, Boston, 1984. MR 86j:22014
  • 25. -, Arithmeticity of holonomy groups of Lie foliations, J. Amer. Math. Soc. 1 (1988), 35-58. MR 89c:22019
  • 26. -, Representations of fundamental groups of manifolds with a semisimple transformation group, J. Amer. Math. Soc. 2 (1989), 201-213. MR 90i:22021

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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

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

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