What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?

Question

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$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 $\PSL(2,\R)$?

Answer 1

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 $\pi_1(S)$.