Conformal great circle flows on the 3-sphere
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- by Adam Harris and Gabriel P. Paternain PDF
- Proc. Amer. Math. Soc. 144 (2016), 1725-1734 Request permission
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
- A. Aazami, The Newman-Penrose formalism for Riemannian 3-manifolds, J. Geom. Phys. (to appear)
- Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
- Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
- 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, DOI 10.1023/A:1013182217545
- Herman Gluck and Frank W. Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983), no. 1, 107–132. MR 700132
- Victor Guillemin, The Radon transform on Zoll surfaces, Advances in Math. 22 (1976), no. 1, 85–119. MR 426063, DOI 10.1016/0001-8708(76)90139-0
- 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, DOI 10.1007/s10455-008-9111-2
- 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, DOI 10.1090/S0002-9947-07-04350-4
- 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
- Gabriel P. Paternain, Geodesic flows, Progress in Mathematics, vol. 180, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1712465, DOI 10.1007/978-1-4612-1600-1
- Yung-chow Wong, Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 589–594. MR 216433, DOI 10.1073/pnas.57.3.589
Additional Information
- Adam Harris
- Affiliation: School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
- MR Author ID: 607698
- Email: adamh@une.edu.au
- Gabriel P. Paternain
- Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom
- Email: g.p.paternain@dpmms.cam.ac.uk
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1725-1734
- MSC (2010): Primary 53D25, 58B20; Secondary 32Q65
- DOI: https://doi.org/10.1090/proc/12819
- MathSciNet review: 3451248