Looking for a criterion for split exact sequence of sheaves of modules similar to Miyata's theorem

Question

Miyata's theorem states that:

For a commutative Noetherian ring R, and a short exact sequence:

$0 \rightarrow M \rightarrow P \rightarrow N \rightarrow 0$

if $P$ is isomorphic to a direct sum of $M$ and $N$, then the sequence split.

Is there a similar criterion for sheaves of $O_X$-modules on a variety? Thanks.