Abstract
The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M → T 1 M, where the unit tangent bundle T 1 M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S 5,S 7,..., and an estimate on the index is obtained.
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References
Eells, J. and Lemaire, L.: Selected Topics in Harmonic Maps, CBMS Regional Conference Series 50, American Math. Soc., Providence, R.I., 1983.
Gromoll, D., Klingenberg,W. and Meyer,W.: Riemannsche Geometrie im Grossen, Lecture Notes in Math. 55, Springer-Verlag, Berlin, 1968.
Gluck, H. and Ziller,W.: On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61(1986), 177–192.
Ishihara, T.: Harmonic sections of tangent bundles, J. Math. Tokushima Univ. 13 (1979), 23–27.
Johnson, D.: Volumes of flows, Proc. Amer. Math. Soc. 104 (1988), 923–931.
Nouhaud, O.: Applications harmoniques d'une vari´et´e riemannienne dans son fibr´e tangent, C.R. Acad. Sci. Paris 284 (1977), 815–818.
Smith, R. T.: The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229–236.
Wood, C. M.: Harmonic sections and Yang-Mills fields, Proc. London Math. Soc. 54 (1987), 544–558.
Wood, C. M.: A class of harmonic almost-product structures, J. Geom. Phys. 14 (1994), 25–42.
Wood, C. M.: Harmonic almost-complex structures, CompositioMath. 99 (1995), 183–212.
Xin, Y-L.: Some results on stable harmonic maps, Duke Math. J. 47 (1980), 609–613.
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Wood, C.M. On the Energy of a Unit Vector Field. Geometriae Dedicata 64, 319–330 (1997). https://doi.org/10.1023/A:1017976425512
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DOI: https://doi.org/10.1023/A:1017976425512