  We investigate the holomorphic differential operators on a Riemann surface
$M$. This is done by endowing $M$ with a projective structure. Let $\mathcal L$
be a theta characteristic on $M$. We explicitly describe the jet bundle
$J^k(E\otimes {\mathcal L}^{\otimes n})$, where $E$ is a holomorphic vector
bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This
provides a description of holomorphic differential operators from $E\otimes
{\mathcal L}^{\otimes n}$ to another holomorphic vector bundle $F$ using the
natural isomorphism $\text{Diff}^k(E\otimes {\mathcal L}^{\otimes n}, F)=
F\otimes (J^k(E\otimes {\mathcal L}^{\otimes n}))^*$.
