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Transactions of the American Mathematical Society

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The tension field of the Gauss map


Authors: Ernst A. Ruh and Jaak Vilms
Journal: Trans. Amer. Math. Soc. 149 (1970), 569-573
MSC: Primary 53.04
MathSciNet review: 0259768
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Abstract: In this paper it is shown that the tension field of the Gauss map can be identified with the covariant derivative of the mean curvature vector field. Since a map with vanishing tension field is called harmonic the following theorem is obtained as a corollary. The Gauss map of a minimal submanifold is harmonic.


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  • [2] James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 0164306
  • [3] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
  • [4] Ernst A. Ruh, Asymptotic behaviour of non-parametric minimal hypersurfaces, J. Differential Geometry 4 (1970), 509–513. MR 0276877

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0259768-5
Keywords: Codazzi equation, energy integral, harmonic map, immersed submanifold, mean curvature vector field, parallel mean curvature, tension field
Article copyright: © Copyright 1970 American Mathematical Society