Rank two vector bundles on \({\mathbb P}^ n\) uniform with respect to some rational curves (Q1098902)

A classical method for studying the classifying vector bundles on projective spaces is to study their restrictions to linear subvarieties. For example, it is well known that a vector bundle on \({\mathbb{P}}^ n \) of rank \(r\leq n,\) whose restriction to all lines has the same splitting type is homogeneous. Let E be a rank two vector bundle on \({\mathbb{P}}^ n,\) whose restriction to all rational smooth curves of degree \(d\) in \({\mathbb{P}}^ n \)has constant splitting type. In this paper we prove that if \(d=2\), or \(d=3\) and \(c_ 1(E)\) is even, then E is homogeneous.