Tohoku Mathematical Journal

Generalized Finsler structures on closed 3-manifolds

Gheorghe Pitiş, Sorin V. Sabau, and Kazuhiro Shibuya

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

An $(I,J,K)$-generalized Finsler structure on a 3-manifold is a generalization of a Finslerian structure, introduced by R. Bryant in order to separate and clarify the local and global aspects in Finsler geometry making use of Cartan's method of exterior differential systems. In this paper, we show that there is a close relation between $(I,J,1)$-generalized Finsler structures and a class of contact circles, namely the so-called Cartan structures.This correspondence allows us to determine the topology of 3-manifolds that admit $(I,J,1)$-generalized Finsler structures and to single out classes of $(I,J,1)$-generalized Finsler structures induced by standard Cartan structures.

Article information

Source
Tohoku Math. J. (2), Volume 66, Number 3 (2014), 321-353.

Dates
First available in Project Euclid: 8 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1412783202

Digital Object Identifier
doi:10.2748/tmj/1412783202

Mathematical Reviews number (MathSciNet)
MR3266736

Zentralblatt MATH identifier
1318.53081

Subjects
Primary: 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20]
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]

Keywords
Generalized Finsler manifolds taut contact circles contact topology

Citation

Sabau, Sorin V.; Shibuya, Kazuhiro; Pitiş, Gheorghe. Generalized Finsler structures on closed 3-manifolds. Tohoku Math. J. (2) 66 (2014), no. 3, 321--353. doi:10.2748/tmj/1412783202. https://projecteuclid.org/euclid.tmj/1412783202


Export citation

References

  • D. Bao, S. S. Chern and Z. Shen, An introduction to Riemann Finsler geometry, Springer, GTM 200, 2000.
  • A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb. 93. Springer-Verlag, Berlin-New York, 1978.
  • R. Bryant, Finsler structures on the 2-sphere satisfying $K=1$, Finsler geometry (Seatle, WA, 1995), 27–41, Contemp. Math., 196, Amer. Math. Soc., Providence, RI, 1996.
  • R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 28 (2002), no. 2, 221–262.
  • H. Geiges and J. Gonzalo, Contact Geometry and Complex Surfaces, Invent. Math. 121 (1995), 147–209.
  • H. Geiges and J. Gonzalo, Contact circles on 3-manifolds, J. Differential Geom. 46 (1997), 236–286.
  • H. Geiges and J. Gonzalo, Moduli of contact circles, J. Reine Angew. Math. 551 (2002), 41–85.
  • J. Gonzalo, Branched covers and contact structures, Proc. Amer. Math. Soc. 101 (1987), 347–352.
  • Th. A. Ivey and L. M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, Grad. Stud. Math. 61. American Mathematical Society, Providence, RI, 2003.
  • S. V. Sabau, K. Shibuya and H. Shimada, On the existence of generalized unicorns on surfaces, Differential Geom. Appl. 28 (2010), 406–435.
  • S. V. Sabau, K. Shibuya and H. Shimada, Moving frames on generalized Finsler structures, J. Korean Math. Soc. 49 (2012), 1229–1257.
  • P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.
  • J. A. Wolf, Spaces of constant curvature. Fifth edition. Publish or Perish, Inc., Houston, TX, 1984.