Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. 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.

Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. They classify such splittings on compact Kaehler manifolds. Those which admit a vector bundle with splitting jet sequence are precisely projective spaces, compact complex manifolds covered by a complex torus, and those compact Kaehler manifolds whose universal covering space is the unit ball in complex Euclidean space. In particular, every complete curve has such a sequence. On the other hand, for splitting of jet sequences where the vector bundle is $\mathcal{O}$, the compact Kaehler manifold is covered by a complex torus. In particular, the only complete curves with such splittings are genus one curves.

Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. They classify such splittings on compact Kaehler manifolds.

Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. 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.