$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(2)$ depending on orientability or a hyperbolic structure as a special $(X,G)$-structure $\pi_1(S)\subset \PSL(2,\R)$. (An $(X,G)$-structure could be regarded as a flat $X$-bundle with a section transversal to the flat connection, so holonomy of the flat bundle is the holonomy of the structure.)

So there are two ways of describing the holonomy of a hyperbolic structure. But, are they both equally as valid? And what is the relationship between $\SO(2)$ and $\PSL(2,\R)$?

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