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 $g = 1$.