Connections which are harmonic with respect to general natural metrics
Let \((M,g)\) be a Riemannian manifold with Levi-Cività connection \(\nabla\). Following [\textit{V. Oproiu}, Math. J. Toyama Univ. 22, 1--14 (1999; Zbl 0955.53039)], one defines a \(6\)-parameter family of pseudo-Riemannian metrics on the tangent bundle \(TM\), called general natural lifted (g.n.l.) metrics. More precisely, a g.n.l. metric \(G\) is specified by six smooth functions \(c_i, d_i: [0,+\infty[\to\mathbb{R}\), \(i\in\{1,2,3\}\), according to the conditions: NEWLINE\[NEWLINE\begin{aligned} G_y(X^h_y, Y^h_y) &= c_1(t) g_{\pi(y)}(X, Y)+ d_1(t) g_{\pi(y)}(X, y) g_{\pi(y)}(Y, y),\\ G_y(X^v_y, Y^v_y) &= c_2(t) g_{\pi(y)}(X, Y)+ d_2(t) g_{\pi(y)}(X, y) g_{\pi(y)}(Y, y),\\ G(X^v_y, Y^h_y) &= G(X^h_y, Y^v_y)= c_3(t) g_{\pi(y)}(X, Y)+ d_3(t) g_{\pi(y)}(X, y) g_{\pi(y)}(Y, y),\end{aligned}NEWLINE\]NEWLINE for any \(y\in TM\), where \(t={1\over 2} g_{\pi(y)}(y,y)\) is the energy density of \(y\) and \(\pi:TM\to M\) the canonical projection. For any vector field \(X\) on \(M\), \(X^h\) (resp. \(X^v\)) denotes the \(\nabla\)-horizontal (resp. vertical) lift of \(X\) to \(TM\).NEWLINENEWLINE A (non)-linear connection on \(M\) is viewed as an almost product structure on \(TM\) whose \(-1\)-eigendistribution coincides with the vertical distribution \(VTM\).NEWLINENEWLINE This paper deals with the harmonicity of any (non)-linear connection with respect to all g.n.l. metrics on \(TM\). More precisely, a (non)-linear connection \(P\) on \((M,g)\) is said to be harmonic with respect to a g.n.l. metric \(G\) on \(TM\) if the codifferential of \(P\) with respect to the Levi-Cività connection of \(G\) vanishes. Firstly, the authors characterize the harmonicity of a (non)-linear connection. Then, considering a Riemannian space-form \((M(c),g)\), they characterize the g.n.l. metrics with respect to which the Levi-Cività connection of \(g\) is harmonic. In particular, the Levi-Cività connection of \((M(c),g)\) is harmonic with respect to the Sasaki metric on \(TM\). The authors also discuss the harmonicity, with respect to a g.n.l. metric, of the canonical almost tangent structure on \(TM\).
scientific article; zbMATH DE number 6054369
scientific article