Algebraic & Geometric Topology

Lipschitz minimality of Hopf fibrations and Hopf vector fields

Dennis DeTurck, Herman Gluck, and Peter Storm

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Abstract

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.

Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibers as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.

Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 3 (2013), 1369-1412.

Dates
Received: 23 May 2012
Revised: 12 October 2012
Accepted: 22 October 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715587

Digital Object Identifier
doi:10.2140/agt.2013.13.1369

Mathematical Reviews number (MathSciNet)
MR3071130

Zentralblatt MATH identifier
1268.53054

Subjects
Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 55R10: Fiber bundles 55R25: Sphere bundles and vector bundles 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx] 57R25: Vector fields, frame fields 57R35: Differentiable mappings
Secondary: 53C38: Calibrations and calibrated geometries 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Keywords
Riemannian submersion Lipschitz constant Lipschitz minimizer Hopf fibration Hopf vector field Grassmannian

Citation

DeTurck, Dennis; Gluck, Herman; Storm, Peter. Lipschitz minimality of Hopf fibrations and Hopf vector fields. Algebr. Geom. Topol. 13 (2013), no. 3, 1369--1412. doi:10.2140/agt.2013.13.1369. https://projecteuclid.org/euclid.agt/1513715587


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References

  • V Borrelli, F Brito, O Gil-Medrano, The infimum of the energy of unit vector fields on odd-dimensional spheres, Ann. Global Anal. Geom. 23 (2003) 129–140
    Mathematical Reviews (MathSciNet): MR1961372
    Zentralblatt MATH: 1031.53090
    Digital Object Identifier: doi:10.1023/A:1022404728764
  • V Borrelli, O Gil-Medrano, A critical radius for unit Hopf vector fields on spheres, Math. Ann. 334 (2006) 731–751
    Mathematical Reviews (MathSciNet): MR2209254
    Zentralblatt MATH: 1115.53025
    Digital Object Identifier: doi:10.1007/s00208-005-0692-9
  • R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer, New York (1982)
    Mathematical Reviews (MathSciNet): MR658304
    Zentralblatt MATH: 0496.55001
  • F G B Brito, Total bending of flows with mean curvature correction, Differential Geom. Appl. 12 (2000) 157–163
    Mathematical Reviews (MathSciNet): MR1758847
    Zentralblatt MATH: 0995.53023
    Digital Object Identifier: doi:10.1016/S0926-2245(00)00007-3
  • F G B Brito, P Walczak, On the energy of unit vector fields with isolated singularities, Ann. Polon. Math. 73 (2000) 269–274
    Mathematical Reviews (MathSciNet): MR1785691
    Zentralblatt MATH: 0997.53025
    Digital Object Identifier: doi:10.4064/ap-73-3-269-274
  • J Eells, L Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1–68
    Mathematical Reviews (MathSciNet): MR495450
    Zentralblatt MATH: 0401.58003
    Digital Object Identifier: doi:10.1112/blms/10.1.1
  • R H Escobales, Jr, Riemannian submersions with totally geodesic fibers, J. Differential Geom. 10 (1975) 253–276
    Mathematical Reviews (MathSciNet): MR0370423
    Zentralblatt MATH: 0301.53024
    Digital Object Identifier: doi:10.4310/jdg/1214432793
    Project Euclid: euclid.jdg/1214432793
  • F B Fuller, Harmonic mappings, Proc. Nat. Acad. Sci. U. S. A. 40 (1954) 987–991
    Mathematical Reviews (MathSciNet): MR0064453
    Zentralblatt MATH: 0056.41102
    Digital Object Identifier: doi:10.1073/pnas.40.10.987
  • O Gil-Medrano, Relationship between volume and energy of vector fields, Differential Geom. Appl. 15 (2001) 137–152
    Mathematical Reviews (MathSciNet): MR1857559
    Zentralblatt MATH: 1066.53068
    Digital Object Identifier: doi:10.1016/S0926-2245(01)00053-5
  • O Gil-Medrano, Volume and energy of vector fields on spheres. A survey, from: “Differential geometry, Valencia, 2001”, (O Gil-Medrano, V Miquel, editors), World Sci. Publ., River Edge, NJ (2002) 167–178
    Mathematical Reviews (MathSciNet): MR1922046
    Zentralblatt MATH: 1029.53074
  • O Gil-Medrano, Unit vector fields that are critical points of the volume and of the energy: characterization and examples, from: “Complex, contact and symmetric manifolds”, (O Kowalski, E Musso, editors), Progr. Math. 234, Birkhäuser, Boston, MA (2005) 165–186
    Mathematical Reviews (MathSciNet): MR2105148
  • O Gil-Medrano, E Llinares-Fuster, Minimal unit vector fields, Tohoku Math. J. 54 (2002) 71–84
    Mathematical Reviews (MathSciNet): MR1878928
    Zentralblatt MATH: 1006.53053
    Digital Object Identifier: doi:10.2748/tmj/1113247180
    Project Euclid: euclid.tmj/1113247180
  • H Gluck, F W Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983) 107–132
    Mathematical Reviews (MathSciNet): MR700132
    Zentralblatt MATH: 0523.55020
    Digital Object Identifier: doi:10.1215/S0012-7094-83-05003-2
    Project Euclid: euclid.dmj/1077303001
  • H Gluck, F Warner, W Ziller, The geometry of the Hopf fibrations, Enseign. Math. 32 (1986) 173–198
    Mathematical Reviews (MathSciNet): MR874686
    Zentralblatt MATH: 0616.53038
  • H Gluck, F Warner, W Ziller, Fibrations of spheres by parallel great spheres and Berger's rigidity theorem, Ann. Global Anal. Geom. 5 (1987) 53–82
    Mathematical Reviews (MathSciNet): MR933907
    Zentralblatt MATH: 0642.53046
    Digital Object Identifier: doi:10.1007/BF00140754
  • H Gluck, W Ziller, On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61 (1986) 177–192
    Mathematical Reviews (MathSciNet): MR856085
    Zentralblatt MATH: 0605.53022
    Digital Object Identifier: doi:10.1007/BF02621910
  • D Gromoll, K Grove, One-dimensional metric foliations in constant curvature spaces, from: “Differential geometry and complex analysis”, (I Chavel, H M Farkas, editors), Springer, Berlin (1985) 165–168
    Mathematical Reviews (MathSciNet): MR780042
    Zentralblatt MATH: 0566.53035
  • D Gromoll, K Grove, The low-dimensional metric foliations of Euclidean spheres, J. Differential Geom. 28 (1988) 143–156
    Mathematical Reviews (MathSciNet): MR950559
    Zentralblatt MATH: 0683.53030
    Digital Object Identifier: doi:10.4310/jdg/1214442164
    Project Euclid: euclid.jdg/1214442164
  • D-S Han, J-W Yim, Unit vector fields on spheres, which are harmonic maps, Math. Z. 227 (1998) 83–92
    Mathematical Reviews (MathSciNet): MR1605381
    Zentralblatt MATH: 0891.53024
    Digital Object Identifier: doi:10.1007/PL00004369
  • H Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931) 637–665
    Mathematical Reviews (MathSciNet): MR1512691
    Digital Object Identifier: doi:10.1007/BF01457962
  • H Hopf, Uber die Abbildungen von Sph"aren auf Sph"aren niedrigerer Dimension, Fund. Math. 25 (1935) 427–440
  • T Ishihara, Harmonic sections of tangent bundles, J. Math. Tokushima Univ. 13 (1979) 23–27
    Mathematical Reviews (MathSciNet): MR563393
    Zentralblatt MATH: 0427.53019
  • D L Johnson, Volumes of flows, Proc. Amer. Math. Soc. 104 (1988) 923–931
    Mathematical Reviews (MathSciNet): MR964875
    Zentralblatt MATH: 0687.58031
    Digital Object Identifier: doi:10.1090/S0002-9939-1988-0964875-4
  • J J Konderak, On harmonic vector fields, Publ. Mat. 36 (1992) 217–228
    Mathematical Reviews (MathSciNet): MR1179614
    Zentralblatt MATH: 0755.53029
    Digital Object Identifier: doi:10.5565/PUBLMAT_36192_17
  • O Nouhaud, Applications harmoniques d'une variété riemannienne dans son fibré tangent. Généralisation, C. R. Acad. Sci. Paris Sér. A-B 284 (1977) A815–A818
    Mathematical Reviews (MathSciNet): MR0431035
  • S L Pedersen, Volumes of vector fields on spheres, Trans. Amer. Math. Soc. 336 (1993) 69–78
    Mathematical Reviews (MathSciNet): MR1079056
    Zentralblatt MATH: 0771.53023
    Digital Object Identifier: doi:10.1090/S0002-9947-1993-1079056-2
  • A Ranjan, Riemannian submersions of spheres with totally geodesic fibres, Osaka J. Math. 22 (1985) 243–260
    Mathematical Reviews (MathSciNet): MR800969
    Zentralblatt MATH: 0576.53035
    Project Euclid: euclid.ojm/1200778256
  • S Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. 10 (1958) 338–354
    Mathematical Reviews (MathSciNet): MR0112152
    Digital Object Identifier: doi:10.2748/tmj/1178244668
    Project Euclid: euclid.tmj/1178244668
  • S W Wei, Classification of stable currents in the product of spheres, Tamkang J. Math. 42 (2011) 427–440
    Mathematical Reviews (MathSciNet): MR2862346
    Zentralblatt MATH: 1257.49048
    Digital Object Identifier: doi:10.5556/j.tkjm.42.2011.427-440
  • J H C Whitehead, An expression of Hopf's invariant as an integral, Proc. Nat. Acad. Sci. U. S. A. 33 (1947) 117–123
    Mathematical Reviews (MathSciNet): MR0020255
    Zentralblatt MATH: 0030.07902
    Digital Object Identifier: doi:10.1073/pnas.33.5.117
  • B Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001) 281–295
    Mathematical Reviews (MathSciNet): MR1826371
    Zentralblatt MATH: 1028.53044
    Digital Object Identifier: doi:10.1007/PL00005801
  • J A Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math. 7 (1963) 447–462
    Mathematical Reviews (MathSciNet): MR0156295
    Zentralblatt MATH: 0114.37101
    Project Euclid: euclid.ijm/1255644952
  • J A Wolf, Geodesic spheres in Grassmann manifolds, Illinois J. Math. 7 (1963) 425–446
    Mathematical Reviews (MathSciNet): MR0156294
    Zentralblatt MATH: 0114.37002
    Project Euclid: euclid.ijm/1255644951
  • Y-C Wong, Isoclinic $n$–planes in Euclidean $2n$–space, Clifford parallels in elliptic $(2n-1)$–space, and the Hurwitz matrix equations, Mem. Amer. Math. Soc. No. 41 (1961)
    Mathematical Reviews (MathSciNet): MR0145437
  • C M Wood, On the energy of a unit vector field, Geom. Dedicata 64 (1997) 319–330
    Mathematical Reviews (MathSciNet): MR1440565
    Zentralblatt MATH: 0878.58017
    Digital Object Identifier: doi:10.1023/A:1017976425512
  • C M Wood, The energy of Hopf vector fields, Manuscripta Math. 101 (2000) 71–88
    Mathematical Reviews (MathSciNet): MR1737225
    Zentralblatt MATH: 0946.53032
    Digital Object Identifier: doi:10.1007/s002290050005
  • Y L Xin, Some results on stable harmonic maps, Duke Math. J. 47 (1980) 609–613
    Mathematical Reviews (MathSciNet): MR587168
    Digital Object Identifier: doi:10.1215/S0012-7094-80-04736-5
    Project Euclid: euclid.dmj/1077314183

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