  We investigate the symplectic geometric and differential geometric aspects of
the moduli space of connections on a compact Riemann surface $X$. Fix a theta
characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli
space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero.
Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we
construct a natural holomorphic connection on $E$ that depends holomorphically
on $E$. Using this holomorphic connection, we construct a canonical holomorphic
isomorphism between the following two: \begin{enumerate} \item the moduli space
$\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a
holomorphic connection on $E$, and
  \item the space ${\rm Conn}(\Theta)$ given by the sheaf of holomorphic
connections on the line bundle on $\mathcal M$ associated to the theta divisor.
\end{enumerate} The above isomorphism between $\mathcal C$ and ${\rm
Conn}(\Theta)$ is symplectic structure preserving, and it moves holomorphically
as $X$ runs over a holomorphic family of Riemann surfaces.
