Killing vector fields on tangent sphere bundles
Let \((TM,g_s)\) be the tangent bundle of a Riemannian manifold \((M,g)\) where \(g_S\) denotes the Sasaki metric and let \((T_\lambda M, \overline{g}_S)\) be the tangent sphere bundle, i.e., the set of elements \((x,u)\) where \(x\in M\) and \(u\in T_x M\), \(g(u,u)= \lambda^2\), endowed with the metric \(\overline{g}_S\) induced from \(g_S\). \(T_\lambda M\) is a hypersurface of \(TM\). S. Tanno has determined the general form of the Killing vector fields on \((TM, g_S)\). In the paper under review, the author uses this result to determine the general form of the Killing vector fields \(X\) on \((T_\lambda M, \overline{g}_S)\) which are fiber-preserving, i.e., those \(X\) such that the (local) isometries \(\psi_t\) generated by \(X\) map each fiber into a fiber. To do this, he proves that each fiber-preserving Killing vector field on \((T_\lambda M, \overline{g}_S)\) can be extended to a Killing vector field on \((TM,g_S)\) and that conversely, the restriction of every Killing vector field on \((TM,g_S)\) which is tangent to \(T_\lambda M\) is a fiber-preserving Killing vector field on \((T_\lambda M,\overline{g}_S)\). The author also provides examples of spaces of constant curvature \((M(c),g)\), with specific curvature \(c\), all of whose Killing vector fields on the unit tangent sphere bundle \((T_1 M, \overline{g}_S)\) are fiber-preserving. This result has already been extended by S. Tanno and the author to all values of \(c\), except for \(c\neq 1\). For this last case, the geodesic flow vector is a Killing vector field on \((T_1 M, \overline{g}_S)\) which is not fiber-preserving.
scientific article; zbMATH DE number 1208196