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Harmonic maps defined by the geodesic flow. (English) Zbl 1251.53038
The Riemannian geometry of the tangent bundle $$TM$$ and the tangent sphere bundle $$T_1M$$ of a Riemannian manifold $$(M, g)$$ started with the works of Sasaki [S. Sasaki, Tôhoku Math. J., II. Ser. 10, 338-354 (1958; Zbl 0086.15003); Tôhoku Math. J., II. Ser. 14, 146–155 (1962; Zbl 0109.40505)], where the metrics used on $$TM$$ and $$T_1M$$ are the so-call Sasaki metric $$g^S$$ and the induced Sasaki metric ($$T_1M$$ is viewed as a hypersurface of $$TM$$). Later in the 1970’s another metric called Cheeger-Gromoll metric $$g^{CG}$$ on $$TM$$ was suggested in [J. Cheeger and D. Gromoll, Ann. Math. (2) 96, 413–443 (1972; Zbl 0246.53049)] and was later constructed explicitly in [E. Musso and F. Tricerri, Ann. Mat. Pura Appl., IV. Ser. 150, 1–19 (1988; Zbl 0658.53045)]. Recently, a family of $$g$$-natural Riemannian metrics on $$TM$$ was introduced and studied (see, e.g., [K. M. T. Abbassi and M. Sarih, Differ. Geom. Appl. 22, No. 1, 19–47 (2005; Zbl 1068.53016); Arch. Math., Brno 41, No. 1, 71–92 (2005; Zbl 1114.53015)]). This 6-parameter family of Riemannian metrics includes the Sasaki metric and the Cheeger-Gromoll metric as members and can be described as $\begin{cases} G(x,u)(X^h,Y^h) = (\alpha_1 + \alpha_3)(r^2)g_x(X, Y )+ (\beta_1 +\beta_3)(r^2)g_x(X, u)g_x(Y, u),\\ G(x,u)(X^h,Y^v) = \alpha_2(r^2)g_x(X, Y )+ \beta_2 (r^2)g_x(X, u)g_x(Y, u),\\ G(x,u)(X^v,Y^v) = \alpha_1(r^2)g_x(X, Y )+ \beta_1 (r^2)g_x(X, u)g_x(Y, u), \end{cases}$ where $$r^2 = g_x(u, u)$$.
The harmonicity of the map $$V:(M, g)\longrightarrow (T_1M, g^S)$$ defined by a unit vector field on $$M$$ has been studied by several authors including G. Wiegmink [Math. Ann. 303, No. 2, 325–344 (1995; Zbl 0834.53034)], C. M. Wood [Geom. Dedicata 64, No. 3, 319–330 (1997; Zbl 0878.58017)], and D. -S. Han and J. -W. Yim [Math. Z. 227, No. 1, 83–92 (1998; Zbl 0891.53024)]. If the base manifold $$(M,g)$$ is two-point homogeneous, then it was proved by E. Boeckx and L. Vanhecke [Differ. Geom. Appl. 13, No. 1, 77–93 (2000; Zbl 0973.53053)] that the map $$\bar\xi: (T_1M, g^S)\longrightarrow (T_1T_1M, (g^S)^S)$$ defined by any geodesic flow vector field is harmonic.
Replacing the Sasaki metric $$g^S$$ by a $$g$$-natural Riemannian metric $$\tilde{G}$$ on the tangent sphere bundle the authors of the paper under review prove that the map $$\bar\xi: (T_1M, \tilde{G})\longrightarrow (T_{\rho}T_1M, \tilde{\tilde{G}})$$ is always a harmonic vector field, i.e., a critical point of the energy restricted to the set of smooth unit vector fields. They also find a necessary and sufficient condition for such a map to be harmonic. Some explicit examples of such maps are also given.

##### MSC:
 53C43 Differential geometric aspects of harmonic maps 53D25 Geodesic flows in symplectic geometry and contact geometry 58E20 Harmonic maps, etc.
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