Harmonic maps defined by the geodesic flow.

*(English)*Zbl 1251.53038The 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.

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.

Reviewer: Ye-Lin Ou (Commerce)