# Article

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Keywords:
geodesic flow; two-point homogeneous spaces; harmonic maps; stability; energy functional
Summary:
We study the stability of the geodesic flow $\xi$ as a critical point for the energy functional when the base space is a compact orientable quotient of a two-point homogeneous space.
References:
[1] Boeckx E., Vanhecke L.: Characteristic reflections on unit tangent sphere bundles. Houston J. Math. 23 (1997), 427-448. MR 1690045 | Zbl 0897.53010
[2] Boeckx E., Vanhecke L.: Harmonic and minimal vector fields on tangent and unit tangent bundles. Differential Geom. Appl. 13 (2000), 77-93. MR 1775222 | Zbl 0973.53053
[3] Borel A.: Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963), 111-122. MR 0146301 | Zbl 0116.38603
[4] Brito F.: Total bending of flows with mean curvature correction. Differential Geom. Appl. 12 (2000), 157-163. MR 1758847 | Zbl 0995.53023
[5] Chen B.Y., Vanhecke L.: Differential geometry of geodesic spheres. J. Reine Angew. Math. 25 (1981), 28-67. MR 0618545 | Zbl 0503.53013
[6] González-Dávila J.C., Vanhecke L.: Energy and volume of unit vector fields on three-dimensional Riemannian manifolds. Differential Geom. Appl., to appear. MR 1900746
[7] Gray A., Vanhecke L.: Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142 (1979), 157-198. MR 0521460 | Zbl 0428.53017
[8] Higuchi A., Kay B.S., Wood C.M.: The energy of unit vector fields on the $3$-sphere. J. Geom. Phys. 37 (2002), 137-155. MR 1807086
[9] Milnor J.: Curvature of left invariant metrics on Lie groups. Adv. in Math. 21 (1976), 293-329. MR 0425012
[10] Tricerri F., Vanhecke L.: Homogeneous Structures on Riemannian Manifolds. Lecture Note Series London Math. Soc. 83, Cambridge Univ. Press, 1983. MR 0712664 | Zbl 0641.53047
[11] Watanabe Y.: Integral inequalities in compact orientable manifolds, Riemannian or Kählerian. Kōdai Math. Sem. Rep. 20 (1968), 264-271. MR 0248702
[12] Wiegmink G.: Total bending of vector fields on Riemannian manifolds. Math. Ann. 303 (1995), 325-344. MR 1348803 | Zbl 0834.53034
[13] Wiegmink G.: Total bending of vector fields on the sphere $S^3$. Differential Geom. Appl. 6 (1996), 219-236. MR 1408308
[14] Wood C.M.: On the energy of a unit vector field. Geom. Dedicata 64 (1997), 319-330. MR 1440565 | Zbl 0878.58017

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