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$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$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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    $\begingroup$ $SO(2)$ is isomorphic to a subgroup of $PSL(2,R)$. $\endgroup$ Apr 30, 2021 at 19:38
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    $\begingroup$ What do you mean by two ways? Do you mean two possible groups: $S0(2)$ or $O(2)$? What do you mean by "equally as valid"? Surely both possibilities occur on different hyperbolic surfaces, but only one or the other on a given hyperbolic surface. $\endgroup$
    – Ben McKay
    Apr 30, 2021 at 19:43
  • $\begingroup$ Well if there an orientability condition, then O(n) has a reduction to SO(n), so its either SO(n) or O(n) depending on if the space is orientable. And equally valid refers to the idea that in differential geometry, you would use either SO(n) and O(n) (depending on Orientability) or in terms of hyperbolic geometry you would use a special $(X,G)$ structure $\pi_1(S) ⊂ PSL(2,R)$. I wanted to know if there was a difference in outcome based on the method used. $\endgroup$ Apr 30, 2021 at 19:52
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    $\begingroup$ @LSpice: They are isomorphic! $\endgroup$ Apr 30, 2021 at 22:37
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    $\begingroup$ Cross-posted at MSE (where this question truly belongs). $\endgroup$ May 1, 2021 at 5:28

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Consider $Q(x_0,x_1,x_2)=x_0^2-x_1^2-x_2^2$, and $H=\{(x_0,x_1,x_2):x_0>0,Q(x_1,x_2,x_3)=1$.

The restriction of $B(x,y)={1\over 2}(Q(x+v)-Q(v))$ to the tangent space of elements of $H$ defines on $H$ a Riemannian metric whose curvature is $-1$. Its group of isometries is the restriction of $O(1,n)$ to $H$.

A (complete) closed Hyperbolic surface $S$ is the quotient of $H$ by a subgroup of isometries $h:\pi_1(S)\rightarrow O(1,n)$, and the holonomy of this structure is defined $h$ ( the holonomy of the flat bundle defined by $h$ over $S$). If the surface is oriented, the image of $h$ is contained in $SO^+(1,n)$ which is isomorphic to $PSL(2,\mathbb{R})$.

There is a natural local diffeomorphism $H\rightarrow \mathbb{R}P^2$ defined by $D(x)=[x]$ where $[x]$ is the projective line throught $x$. $D(x)$ is the developing map of the projective structure defined on $S$ the quotient of $H$ by $\pi_1(S)$.

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