A rational obstruction to be a Gottlieb map

We investigate {\it Gottlieb map}s, which are maps $f:E\to B$ that induce the maps between the Gottlieb groups $\pi_n (f)|_{G_n(E)}:G_n(E)\to G_n(B)$ for all $n$, from a rational homotopy theory point of view.We will define the obstruction group $O(f)$ to be a Gottlieb map and a numerical invariant $o(f)$. It naturally deduces a relative splitting of $E$ in certain cases. We also illustrate several rational examples of Gottlieb maps and non-Gottlieb maps by using derivation arguments in Sullivan models.