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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.

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