On the triviality of direct image of vector bundles

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of smooth projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $\pi$ such that there exists a vector bundle $E$ on $X$ whose direct image $\pi_*E$ is trivial. We show that the existence of $E$ is guided by the properties of the branching divisor of $\pi$. When the covering $\pi\,:\, X \,\longrightarrow\, Y$ is ramified abelian Galois, we give a complete answer. As an application, we prove every smooth ramified abelian Galois covering of $\mathbb{P}^n$ supports an Ulrich bundle.