Harmonic metrics, harmonic tensors, and Gauss maps

Bang-yen CHEN; Tadashi NAGANO

doi:10.2969/jmsj/03620295

Bang-yen CHEN, Tadashi NAGANO

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1984 Volume 36 Issue 2 Pages 295-313

DOI https://doi.org/10.2969/jmsj/03620295

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Published: 1984 Received: May 04, 1982 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry, 3 (1969), 379-392.
2) B. Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973.
3) B. Y. Chen and S. Yamaguchi, Classification of surfaces with totally geodesic Gauss image, Indiana Univ. Math. J., 32 (1983), 143-154.
4) J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68.
5) J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer, J. Math., 86 (1964), 109-160.
6) W. B. Gordon, Convex functions and harmonic maps, Proc. Amer. Math. Soc., 33 (1972), 433-437.
7) R. Harvey and H. B. Lawson, A constellation of minimal varieties defined over the group G₂, Partial Differential Equations and Geometry (C. I. Byrnes, ed.), Dekker, New York, 1979.
8) S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Univ. Press, 1973.
9) S. Kobayashi and K. Nomizu, Foundation of Differential Geometry, I, II, Interscience, New York, 1963, 1969.
10) M. Obata, The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature, J. Differential Geometry, 2 (1968), 217-223.
11) E. A. Ruh and J. Vilms, The tensor field of the Gauss map, Trans. Amer. Math. Soc., 149 (1970), 569-573.
12) J.H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. Ecole Norm. Sup., 11 (1978), 211-228.
13) R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc., 47 (1975), 229-236.
14) K. Yano, The Theory of Lie Derivatives and its Applications, North Holland, Amsterdam, 1957.
15) K. Yano and T. Nagano, On geodesic vector fields in a compact orientable Riemannian space, Comm. Math. Helv., 35 (1961), 55-64.
16) K. Yano and T. Nagano, Les champs des vecteurs géodesiques sur les espaces symétriques, C. R. Acad. Sci. Paris, 252 (1961), 504-505.

Right : [1] M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry, 3 (1969), 379-392.
[2] B. Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973.
[3] B. Y. Chen and S. Yamaguchi, Classification of surfaces with totally geodesic Gauss image, Indiana Univ. Math. J., 32 (1983), 143-154.
[4] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68.
[5] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer, J. Math., 86 (1964), 109-160.
[6] W. B. Gordon, Convex functions and harmonic maps, Proc. Amer. Math. Soc., 33 (1972), 433-437.
[7] R. Harvey and H. B. Lawson, A constellation of minimal varieties defined over the group G₂, Partial Differential Equations and Geometry (C. I. Byrnes, ed.), Dekker, New York, 1979.
[8] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Univ. Press, 1973.
[9] S. Kobayashi and K. Nomizu, Foundation of Differential Geometry, I, II, Interscience, New York, 1963, 1969.
[10] M. Obata, The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature, J. Differential Geometry, 2 (1968), 217-223.
[11] E. A. Ruh and J. Vilms, The tensor field of the Gauss map, Trans. Amer. Math. Soc., 149 (1970), 569-573.
[12] J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. Ecole Norm. Sup., 11 (1978), 211-228.
[13] R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc., 47 (1975), 229-236.
[14] K. Yano, The Theory of Lie Derivatives and its Applications, North Holland, Amsterdam, 1957.
[15] K. Yano and T. Nagano, On geodesic vector fields in a compact orientable Riemannian space, Comm. Math. Helv., 35 (1961), 55-64.
[16] K. Yano and T. Nagano, Les champs des vecteurs géodesiques sur les espaces symétriques, C. R. Acad. Sci. Paris, 252 (1961), 504-505.

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