Let $X$ be a smooth variety. Because $\mathcal{O}\_X$ admits a canonical connection $\mathrm{d} : \mathcal{O}\_X \to \Omega\_X$ the sequence,
$$ 0 \to \Omega\_X \to J^1(\mathcal{O}\_X) \to \mathcal{O}\_X \to 0$$
splits canonically.


I say that $X$ has the jet splitting property at level $n$ if the sequence,
$$ 0 \to \mathrm{Sym}^n(\Omega\_X) \to J^{n}(\mathcal{O}\_X) \to J^{n-1}(\mathcal{O}\_X) \to 0$$
splits (we could say $\mathcal{O}\_X$ admits a $n^{\mathrm{th}}$-order connection in this case).



> 
> **Question.** What is known about varieties with jet splitting at level $n > 1$?
> 
> 
> 


On the affine space, these sequences are always split. For a complete curve, I do not believe there can ever be splitting at any level $n > 1$ unless $g = 1$.



Splitting of higher order jet sequence
Look at the paper P. Jahnke and I. Radloff, [Splitting jet sequences](https://arxiv.org/abs/math/0210454). 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.


