Geodesic
Submanifolds with the harmonic Gauss map in Lie groups
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008) no. 2, pp. 278-293
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In this paper we find a criterion for the Gauss map of an immersed smooth submanifold in some Lie group with left invariant metric to be harmonic. Using the obtained expression we prove some necessary and sufficient conditions for the harmonicity of this map in the case of totally geodesic submanifolds in Lie groups admitting biinvariant metrics. We show that, depending on the structure of the tangent space of a submanifold, the Gauss map can be harmonic in all biinvariant metrics or nonharmonic in some metric. For 2-step nilpotent groups we prove that the Gauss map of a geodesic is harmonic if and only if it is constant. 
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     title = {Submanifolds with the harmonic {Gauss} map in {Lie} groups},
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Ye. V. Petrov. Submanifolds with the harmonic Gauss map in Lie groups. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008) no. 2, pp. 278-293. http://geodesic.mathdoc.fr/item/JMAG_2008_4_2_a4/

[1] P. B. Eberlein, “Geometry of 2-Step Nilpotent Groups with a Left Invariant Metric”, Ann. Sci. École Norm. Sup., 27 (1994), 611–660 | MR | Zbl

[2] P. B. Eberlein, “Geometry of 2-Step Nilpotent Groups with a Left Invariant Metric, II”, Trans. Amer. Math. Soc., 343 (1994), 805–828 | DOI | MR | Zbl

[3] P. B. Eberlein, “Riemannian Submersions and Lattices in 2-Step Nilpotent Lie Groups”, Comm. Analysis and Geom., 11 (2003), 441–488 | DOI | MR | Zbl

[4] N. do Espirito-Santo, S. Fornari, K. Frensel, J. Ripoll, “Constant Mean Curvature Hypersurfaces in a Lie Groups with a Biinvariant Metric”, Manuscripta Math., 111 (2003), 459–470 | DOI | MR | Zbl

[5] S. Kobayashi, “Isometric Imbeddings of Compact Symmetric Spaces”, Tohoku Math. J., 20 (1968), 21–25 | DOI | MR | Zbl

[6] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, v. II, Interscience Publishers John Wiley and Sons, New York, 1969, 470 pp. | MR | Zbl

[7] J. Milnor, “Curvatures of Left Invariant Metrics on Lie Groups”, Adv. Math., 21 (1976), 293–329 | DOI | MR | Zbl

[8] Ye. V. Petrov, “The Gauss Map of Hypersurfaces in 2-Step Nilpotent Lie Groups”, J. Math. Phys., Anal., Geom., 2 (2006), 186–206 | MR | Zbl

[9] E. A. Ruh, J. Vilms, “The Tension Field of the Gauss Map”, Trans. Amer. Math. Soc., 149 (1970), 569–573 | DOI | MR | Zbl

[10] H. Urakawa, Calculus of Variations and Harmonic Maps, Amer. Math. Soc., Providence, RI, 1993, 251 pp. | MR | Zbl