Geometric flow and rigidity on symmetric spaces of noncompact type

Kim, Inkang

doi:10.1090/S0002-9947-00-02566-6

Geometric flow and rigidity on symmetric spaces of noncompact type
HTML articles powered by AMS MathViewer

by Inkang Kim PDF

Trans. Amer. Math. Soc. 352 (2000), 3623-3638 Request permission

Abstract:

In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of $X$, where $X$ is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit.

References

Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. I, Trans. Amer. Math. Soc. 215 (1976), 323–362. MR 394507, DOI 10.1090/S0002-9947-1976-0394507-4
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
M. Druetta, Homogeneous Riemannian manifolds and the visibility axiom, Geom. Dedicata 17, 239-251 (1985).
P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648
F. D. Veldkamp, In honor of Hans Freudenthal on his eightieth birthday, Geom. Dedicata 19 (1985), no. 1, 2–5. MR 797150, DOI 10.1007/BF00233100
J. Harrison, $C^2$ counterexamples to the Seifert conjecture, Topology 27 (1988), no. 3, 249–278. MR 963630, DOI 10.1016/0040-9383(88)90009-2
Jens Heber, On the geometric rank of homogeneous spaces of nonpositive curvature, Invent. Math. 112 (1993), no. 1, 151–170. MR 1207480, DOI 10.1007/BF01232428
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, DOI 10.1007/BF01232679
Y. Kamishima, Geometric Flows on Compact Manifolds and Global Rigidity, Topology 35, 439-450, 1996.
I. Kim, Marked length rigidity of rank one symmetric spaces and their product, to appear in Topology.
Krystyna Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. of Math. (2) 140 (1994), no. 3, 723–732. MR 1307902, DOI 10.2307/2118623
G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
Paul A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math. (2) 100 (1974), 386–400. MR 356086, DOI 10.2307/1971077
W. P. Thurston, Three Dimensional Geometry and Topology, The Geometry Center, unpublished manuscript.