Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Existence of smooth invariant measures for geodesic flows of foliations of Riemannian manifolds

Author: PawełG. Walczak
Journal: Proc. Amer. Math. Soc. 120 (1994), 903-906
MSC: Primary 58F18; Secondary 57R30, 58F17
MathSciNet review: 1227531
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a nontrivial smooth finite measure invariant under the geodesic flow of a foliation $ \mathcal{F}$ of a compact Riemannian manifold $ M$ assuming that the transverse mean curvature of $ \mathcal{F}$ is distributed "nicely" along some leaf geodesics.

References [Enhancements On Off] (What's this?)

  • [CN] César Camacho and Alcides Lins Neto, Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman. MR 824240
  • [G] D. Gabai, Foliations and the topology of $ 3$-manifolds, J. Differential Geom. 18 (1983), 445-503; 26 (1987), 461-478, 479-536.
  • [K] Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR 666697
  • [M] Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254
  • [N] S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 0200938
  • [OS] R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows, Invent. Math. 77 (1984), no. 3, 455–462. MR 759262, 10.1007/BF01388833
  • [P] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
  • [R] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1986.
  • [W] Paweł G. Walczak, Dynamics of the geodesic flow of a foliation, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 637–650. MR 980802, 10.1017/S0143385700004740
  • [Wa] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F18, 57R30, 58F17

Retrieve articles in all journals with MSC: 58F18, 57R30, 58F17

Additional Information

Keywords: Foliation, Riemannian manifold, geodesic flow, invariant measure, entropy
Article copyright: © Copyright 1994 American Mathematical Society