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We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree$0$ over a compact Riemann surface and the affine bundle$\mathscr{A}\to\mathscr{N}$ of flat connections. Following the similaritybetween the Teichm\"{u}ller spaces and the moduli of bundles, we introduce theanalogue of of the quasi-Fuchsian projective connections - local holomorphicsections of $\mathscr{A}$ - that allow to pull back the Liouville symplecticform on $T^{*}\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of theGoldman form to $\mathscr{A}$ by the Riemann-Hilbert correspondence coincideswith the pullback of the Liouville form. We also include a simple proof, in thespirit of Riemann bilinear relations, of the classic result - the pullback ofGoldman symplectic form to $\mathscr{N}$ by the Narasimhan-Seshadri connectionis the natural symplectic form on $\mathscr{N}$, introduced by Narasimhan andAtiyah & Bott.

Goldman form, flat connections and stable vector bundles