SCHWEITZER 1997 Problem 9

Let ( M , g ) be a Riemannian manifold. Extend the metric tensor g to the set of tangents TM with the following specification: if $a,b\in T_v TM (v\in T_p M)$ , then $$\tilde g_v (a, b): = g_p (\dot {\alpha} (0), \dot {\beta} (0) ) + g_p (D _ {\alpha} X(0) , D_{\beta} Y(0) )$$ where $\alpha, \beta$ are curves in M such that $\alpha(0) = \beta(0) = p$ . X and Y are vector fields along $\alpha,\beta$ respectively, with the condition $\dot X (0) = a,\dot Y(0) = b$ . $D _{\alpha}$ and $D _{\beta}$ are the operators of the covariant derivative along the corresponding curves according to the Levi-Civita connection. Is the eigenfunction from the Riemannian manifold (M,g) to the Riemannian manifold $(TM, \tilde g)$ harmonic?