  We propose a conjecture that the monodromy group of a singular hyperbolic
metric on a non-hyperbolic Riemann surface is {\it Zariski dense} in ${\rm
PSL}(2,\,{\Bbb R})$. By using meromorphic differentials and affine connections,
we obtain an evidence of the conjecture that the monodromy group of the
singular hyperbolic metric can not be contained in four classes of
one-dimensional Lie subgroups of ${\rm PSL}(2,\,{\Bbb R})$. Moreover, we
confirm the conjecture if the Riemann surface is either one of the once
punctured Riemann sphere, the twice punctured Riemann sphere, a once punctured
torus and a compact Riemann surface.
