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Conformal great circle flows on the 3-sphere

Authors: Adam Harris and Gabriel P. Paternain
Journal: Proc. Amer. Math. Soc. 144 (2016), 1725-1734
MSC (2010): Primary 53D25, 58B20; Secondary 32Q65
Published electronically: August 12, 2015
MathSciNet review: 3451248
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Abstract: We consider a closed orientable Riemannian 3-manifold $ (M,g)$ and a vector field $ X$ with unit norm whose integral curves are geodesics of $ g$. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of $ g$. We study when this 2-plane bundle remains invariant under two natural almost-complex structures. We also provide a geometric condition that ensures that $ X$ is the Reeb vector field of the 1-form $ \lambda $ obtained by contracting $ g$ with $ X$. We apply these results to the case of great circle flows on the 3-sphere with two objectives in mind: one is to recover the result in a work of Gluck and Gu that a volume-preserving great circle flow must be Hopf and the other is to characterize in a similar fashion great circle flows that are conformal relative to the almost-complex structure in the kernel of $ \lambda $ given by rotation by $ \pi /2$ according to the orientation of $ M$.

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

  • [1] A. Aazami, The Newman-Penrose formalism for Riemannian 3-manifolds, J. Geom. Phys. (to appear)
  • [2] Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062 (2006m:53002)
  • [3] Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738 (2008m:57064)
  • [4] Herman Gluck and Weiqing Gu, Volume-preserving great circle flows on the 3-sphere, Geom. Dedicata 88 (2001), no. 1-3, 259-282. MR 1877220 (2002j:53095),
  • [5] Herman Gluck and Frank W. Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983), no. 1, 107-132. MR 700132 (84g:53056)
  • [6] Victor Guillemin, The Radon transform on Zoll surfaces, Advances in Math. 22 (1976), no. 1, 85-119. MR 0426063 (54 #14009)
  • [7] Adam Harris and Gabriel P. Paternain, Dynamically convex Finsler metrics and $ J$-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom. 34 (2008), no. 2, 115-134. MR 2425525 (2009h:32035),
  • [8] Adam Harris and Krzysztof Wysocki, Branch structure of $ J$-holomorphic curves near periodic orbits of a contact manifold, Trans. Amer. Math. Soc. 360 (2008), no. 4, 2131-2152. MR 2366977 (2008k:32075),
  • [9] R. Mañé, On a theorem of Klingenberg, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985) Pitman Res. Notes Math. Ser., vol. 160, Longman Sci. Tech., Harlow, 1987, pp. 319-345. MR 907897 (88k:58129)
  • [10] Gabriel P. Paternain, Geodesic flows, Progress in Mathematics, vol. 180, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1712465 (2000h:53108)
  • [11] Yung-chow Wong, Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 589-594. MR 0216433 (35 #7266)

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Additional Information

Adam Harris
Affiliation: School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

Gabriel P. Paternain
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom

Received by editor(s): August 11, 2014
Received by editor(s) in revised form: April 28, 2015
Published electronically: August 12, 2015
Communicated by: Guofang Wei
Article copyright: © Copyright 2015 American Mathematical Society

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