On vector bundles over rationally connected varieties

Following the papers by Indranil Biswas and Joao Pedro P. Dos Santos (“On the vector bundles over rationally connected varieties”, “Triviality criteria for vector bundles over rationally connected varieties”) we will prove the following triviality criteria for a vector bundle $E$ on a rationally connected variety $X$: if for every rational curve on $X$ the restriction of $E$ to every rational curve is trivial, then $E$ is itself trivial.