Algebraic & Geometric Topology Volume 13, issue 3 (2013)

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Lipschitz minimality of Hopf fibrations and Hopf vector fields

Dennis DeTurck, Herman Gluck and Peter Storm

Algebraic & Geometric Topology 13 (2013) 1369–1412

DOI: 10.2140/agt.2013.13.1369

Bibliography

1	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 MR1961372
2	V Borrelli, O Gil-Medrano, A critical radius for unit Hopf vector fields on spheres, Math. Ann. 334 (2006) 731 MR2209254
3	R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer (1982) MR658304
4	F G B Brito, Total bending of flows with mean curvature correction, Differential Geom. Appl. 12 (2000) 157 MR1758847
5	F G B Brito, P Walczak, On the energy of unit vector fields with isolated singularities, Ann. Polon. Math. 73 (2000) 269 MR1785691
6	J Eells, L Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1 MR495450
7	R H Escobales Jr., Riemannian submersions with totally geodesic fibers, J. Differential Geom. 10 (1975) 253 MR0370423
8	F B Fuller, Harmonic mappings, Proc. Nat. Acad. Sci. U. S. A. 40 (1954) 987 MR0064453
9	O Gil-Medrano, Relationship between volume and energy of vector fields, Differential Geom. Appl. 15 (2001) 137 MR1857559
10	O Gil-Medrano, Volume and energy of vector fields on spheres. A survey, from: "Differential geometry, Valencia, 2001" (editors O Gil-Medrano, V Miquel), World Sci. Publ., River Edge, NJ (2002) 167 MR1922046
11	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" (editors O Kowalski, E Musso), Progr. Math. 234, Birkhäuser (2005) 165 MR2105148
12	O Gil-Medrano, E Llinares-Fuster, Minimal unit vector fields, Tohoku Math. J. 54 (2002) 71 MR1878928
13	H Gluck, F W Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983) 107 MR700132
14	H Gluck, F Warner, W Ziller, The geometry of the Hopf fibrations, Enseign. Math. 32 (1986) 173 MR874686
15	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 MR933907
16	H Gluck, W Ziller, On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61 (1986) 177 MR856085
17	D Gromoll, K Grove, One-dimensional metric foliations in constant curvature spaces, from: "Differential geometry and complex analysis" (editors I Chavel, H M Farkas), Springer (1985) 165 MR780042
18	D Gromoll, K Grove, The low-dimensional metric foliations of Euclidean spheres, J. Differential Geom. 28 (1988) 143 MR950559
19	D S Han, J W Yim, Unit vector fields on spheres, which are harmonic maps, Math. Z. 227 (1998) 83 MR1605381
20	H Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931) 637 MR1512691
21	H Hopf, Uber die Abbildungen von Sph”aren auf Sph”aren niedrigerer Dimension, Fund. Math. 25 (1935) 427
22	T Ishihara, Harmonic sections of tangent bundles, J. Math. Tokushima Univ. 13 (1979) 23 MR563393
23	D L Johnson, Volumes of flows, Proc. Amer. Math. Soc. 104 (1988) 923 MR964875
24	J J Konderak, On harmonic vector fields, Publ. Mat. 36 (1992) 217 MR1179614
25	O Nouhaud, Applications harmoniques d'une variété riemannienne dans son fibré tangent. Généralisation, C. R. Acad. Sci. Paris Sér. A-B 284 (1977) MR0431035
26	S L Pedersen, Volumes of vector fields on spheres, Trans. Amer. Math. Soc. 336 (1993) 69 MR1079056
27	A Ranjan, Riemannian submersions of spheres with totally geodesic fibres, Osaka J. Math. 22 (1985) 243 MR800969
28	S Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. 10 (1958) 338 MR0112152
29	S W Wei, Classification of stable currents in the product of spheres, Tamkang J. Math. 42 (2011) 427 MR2862346
30	J H C Whitehead, An expression of Hopf's invariant as an integral, Proc. Nat. Acad. Sci. U. S. A. 33 (1947) 117 MR0020255
31	B Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001) 281 MR1826371
32	J A Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math. 7 (1963) 447 MR0156295
33	J A Wolf, Geodesic spheres in Grassmann manifolds, Illinois J. Math. 7 (1963) 425 MR0156294
34	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) MR0145437
35	C M Wood, On the energy of a unit vector field, Geom. Dedicata 64 (1997) 319 MR1440565
36	C M Wood, The energy of Hopf vector fields, Manuscripta Math. 101 (2000) 71 MR1737225
37	Y L Xin, Some results on stable harmonic maps, Duke Math. J. 47 (1980) 609 MR587168