Geodesic
Invariant totally geodesic unit vector fields on three-dimensional Lie groups
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (2007) no. 2, pp. 253-276 Cet article a éte moissonné depuis la source Math-Net.Ru

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We give a complete list of left-invariant unit vector fields on three-dimensional Lie groups equipped with a left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group equipped with the Sasaki metric. As a result we obtain that each three-dimensional Lie group admits totally geodesic unit vector field under some conditions on structural constants. From a geometrical viewpoint, the field is either parallel or a characteristic vector field of a natural almost contact structure on the group. 
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A. Yampolsky. Invariant totally geodesic unit vector fields on three-dimensional Lie groups. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (2007) no. 2, pp. 253-276. http://geodesic.mathdoc.fr/item/JMAG_2007_3_2_a7/

[1] A. Besse, Manifolds All of whose Geodesics are Closed, Springer-Verlag, Berlin–Heidelberg–New York, 1978 | MR | Zbl

[2] Russian Math. Surveys, 46:6 (1991), 55–106 | DOI | MR | Zbl

[3] P. Dombrowski, “On the Geometry of the Tangent Bundle”, J. Reine Angew. Math., 210 (1962), 73–88 | DOI | MR | Zbl

[4] O. Gil-Medrano, “Relationship between Volume and Energy of Unit Vector Fields”, Diff. Geom. Appl., 15 (2001), 137–152 | DOI | MR | Zbl

[5] J. C. González-Dávila, L. Vanhecke, “Examples of Minimal Unit Vector Fields”, Ann. Global Anal. Geom., 18 (2000), 385–404 | DOI | MR | Zbl

[6] J. C. González-Dávila, L. Vanhecke, “Minimal and Harmonic Characteristic Vector Fields on Three-Dimensional Contact Metric Manifolds”, J. Geom., 72 (2001), 65–76 | DOI | MR | Zbl

[7] H. Gluck, W. Ziller, “On the Volume of a Unit Vector Field on the Three-Sphere”, Comm. Math. Helv., 61 (1986), 177–192 | DOI | MR | Zbl

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

[9] K. Tsukada, L. Vanhecke, “Invariant Minimal Unit Vector Field on Lie Groups”, Period. Math. Hungar., 40 (2000), 123–133 | DOI | MR | Zbl

[10] G. Weigmink, “Total Bending of Vector Fields on Riemannian Manifolds”, Math. Ann., 303 (1995), 325–344 | DOI | MR

[11] A. Yampolsky, “On the Mean Curvature of a Unit Vector Field”, Math. Publ. Debrecen, 60:1–2 (2002), 131–155 | MR | Zbl

[12] A. Yampolsky, “A Totally Geodesic Property of Hopf Vector Fields”, Acta Math. Hungar., 101:1–2 (2003), 93–112 | DOI | MR | Zbl

[13] A. Yampolsky, “Full Description of Totally Geodesic Unit Vector Fields on Riemannian 2-Manifold”, Mat. fiz., anal., geom., 11 (2004), 355–365 | MR | Zbl

[14] A. Yampolsky, “On Special Types of Minimal and Totally Geodesic Unit Vector Fields”, Proc. Seventh Int. Conf. Geom., Integrability and Quantization (Bulgaria, Varna, June 2–10, 2005), 292–306 | MR | Zbl