Look at the paper P. Jahnke and I. Radloff, [Splitting jet sequences][1]. They classify such splittings on compact Kaehler manifolds. They are precisely projective spaces and those compact Kaehler manifolds whose universal covering space is complex Euclidean space or the unit ball in complex Euclidean space. In particular, every complete curve has such a sequence.


  [1]: https://arxiv.org/abs/math/0210454