Tohoku Mathematical Journal

Distributions on Riemannian manifolds, which are harmonic maps

Boo-Yong Choi and Jin-Whan Yim

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Abstract

We find new examples of harmaonic maps between compact Riemannian manifolds. A section of a Riemannian fibration is called harmonic if it is harmonic as a map from the base manifold into the total space. When the fibres are totally geodesic, the Euler-Lagrange equation for such sections is formulated. In the case of distributions, which are sections of a Grassmannian bundle, this formula is described in terms of the geometry of base manifolds. Examples of harmonic distributions are constructed when the base manifolds are homogeneous spaces and the integral submanifolds are totally geodesic. In particular, we show all the generalized Hopf-fibrations define harmonic maps into the Grassmannian bundles with the standard metric.

Article information

Source
Tohoku Math. J. (2), Volume 55, Number 2 (2003), 175-188.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246937

Digital Object Identifier
doi:10.2748/tmj/1113246937

Mathematical Reviews number (MathSciNet)
MR1979495

Zentralblatt MATH identifier
1041.53041

Subjects
Primary: 58E20: Harmonic maps [See also 53C43], etc.
Secondary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Keywords
Harmonic map distribution homogeneous space

Citation

Choi, Boo-Yong; Yim, Jin-Whan. Distributions on Riemannian manifolds, which are harmonic maps. Tohoku Math. J. (2) 55 (2003), no. 2, 175--188. doi:10.2748/tmj/1113246937. https://projecteuclid.org/euclid.tmj/1113246937


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References

  • A. L. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987.
  • J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1--68.
  • J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385--524.
  • R. Escobles, Riemannian submersions with totally geodesic fibres, J. Differential. Geom. 10 (1975), 253--276.
  • H. Gluck and W. Gu, Volume-preserving great circle flows on the 3-sphere, Geom. Dedicata 88 (2001), 259--282.
  • D.-S. Han and J.-W. Yim, Unit vector fields on sphere, which are harmonic maps, Math. Z. 227 (1998), 83--92.
  • T. Ishihara, Harmonic section of tangent bundles, J. Math. Tokushima Univ. 13 (1979), 23--27.
  • G. Jensen and M. Rigoli, Harmonic Gauss maps, Pacific J. Math. 136 (1989), 261--282.
  • J. J. Konderak, On harmonic vector field, Publ. Mat. 36 (1992), 217--228.
  • B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459--469.
  • T. Püttmann, Optimal pinching constant of odd dimensional homogeneous spaces, Invent. Math. 138 (1999), 631--684.
  • H. Urakawa, Calculus of Variations and Harmonic Maps, Transl. Math. Monogr, 132, American Mathematical Society, Province, RI, 1993.
  • C. M. Wood, The Gauss section of a Riemannian immersion, J. London Math. Soc. (2) 33 (1986), 157--168.
  • Z. Zhang, Best distributions on Riemannian manifolds and elastic deformations with constant principal strains, Ph. D. thesis, University of Pennsylvania, 1993.