Appendix B Projective structures

We review some basic concepts from projective differential geometry which will motivate the definition of \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) made above as well as provide the natural context for the symmetries of the functionals of (1.2), (1.3) and (1.4). This is a vast subject with many different perspectives and we present only a summarized version. We refer the interested reader to the excellent book [6] by Ovsienko and Tabachnikov as well as their article [7] – these were our main sources for this material.

B.1 One-Dimensional Projective Differential Geometry

Let \(M\) be a one-dimensional oriented manifold.

We fix a square root \((T^*M)^{1/2}\) of the cotangent bundle of \(M\) so that we have an isomorphism of line bundles \[(T^*M)^{1/2}\otimes (T^*M)^{1/2}\simeq T^*M.\]

For an integer \(\ell\) we denote by \(\Omega^{\ell/2}(M)\) the space of smooth densities of weight \(\ell/2\) on \(M\). That is, an element in \(\Omega^{\ell/2}(M)\) is a smooth section of the \(\ell\)-th tensorial power of \((T^*M)^{1/2}\).

As usual, for \(\ell < 0\) we define \[\left((T^*M)^{1/2}\right)^{\otimes \ell}=\left((TM)^{1/2}\right)^{\otimes(-\ell)}\] where \((TM)^{1/2}\) denotes the dual bundle of \((T^*M)^{1/2}\).

Note that an affine connection \(\nabla\) on \(TM\simeq (T^*M)^{-1}\) induces a connection on all tensorial powers of \((T^*M)^{1/2}\). By standard abuse of notation, we will denote these connections by \(\nabla\) as well. In particular, we have first order differential operators \[\nabla: \Omega^{\ell/2}(M)\to \Omega^{\ell/2+1}(M).\] A real projective structure, \(\mathcal{P}\) on \(M\) is a second-order elliptic differential operator \[\mathcal{P}:\Omega^{-1/2}(M)\to \Omega^{3/2}(M)\] so that there is some affine connection \(\nabla\) on \(M\) and \(P\in \Omega^2(M)\) with \[\mathcal{P}=\nabla^2+P.\]

One verifies that, given two real projective structures \(\mathcal{P}_1\) and \(\mathcal{P}_2\), \(\mathcal{P}_2-\mathcal{P}_1\in \Omega^2(M)\) is a zero-order operator. Hence, the space of real projective structures is an affine space with associated vector space \(\Omega^2(M)\). Given an orientation preserving smooth diffeomorphism \(\phi:M_1\to M_2\) we define the push forward and pull back of real projective structures \(\mathcal{P}_i\) on \(M_i\) in an obvious fashion. That is, \[\begin{array}{ccc} (\phi_* \mathcal{P}_1) \cdot \theta = (\phi^{-1})^*\left(\mathcal{P}_1 \cdot \phi^* \theta\right) & \mbox{and} & (\phi^* \mathcal{P}_2) \cdot \theta = \phi^*\left(\mathcal{P}_2 \cdot (\phi^{-1})^* \theta\right). \end{array}\] The Schwarzian derivative of \(\phi\) relative to \(\mathcal{P}_1,\mathcal{P}_2\) is \[S_{\mathcal{P}_1, \mathcal{P}_2}(\phi)= \phi^* \mathcal{P}_2-\mathcal{P}_1 \in \Omega^2(M_1).\] The Schwarzian satisfies the following co-cycle condition \[\tag{B.1} S_{\mathcal{P}_1, \mathcal{P}_3}(\phi_2 \circ \phi_1)=\phi_1^* S_{\mathcal{P}_2,\mathcal{P}_3 }(\phi_2)+ S_{\mathcal{P}_1,\mathcal{P}_2}(\phi_1).\] Given a \(\phi \in \mathrm{ Diff}_+^\infty(M)\) and a real projective structure \(\mathcal{P}\) write \(S_{\mathcal{P}} (\phi)=S_{\mathcal{P}, \mathcal{P}}(\phi)\). An orientation preserving diffeomorphism \(\phi\) is a Möbius transformation of \(\mathcal{P}\) if and only if \(S_{\mathcal{P}}(\phi)=0\). The co-cycle condition implies that the set of such maps forms a subgroup, \(\mathrm{ M\ddot{o}b}(\mathcal{P})\), of \(\mathrm{ Diff}_+^\infty(M)\).

Let \(\mathbb{RP}^1\) be the one-dimensional real projective space – in other words the space of unoriented lines through the origin in \(\mathbb R^2\). Let \((x_1, x_2)\) be the usual linear coordinates on \(\mathbb R^2\). If \((x_1,x_2)\neq 0\), then we denote by \([x_1 : x_2]\) the point in \(\mathbb{RP}^1\) corresponding to the line through the origin and \((x_1,x_2)\). On the chart \(U=\left\{[x_1,x_2]: x_2\neq 0\right\}\) we have the affine coordinate \(\tau=x_1/x_2\) for \(\mathbb{RP}^1\). Let \({}\tau\nabla\) be the (unique) connection so that \(\partial_\tau\) is parallel. There is a unique real projective structure \(\mathcal{P}_{\mathbb{RP}^1}\) on \(\mathbb{RP}^1\) so that \(\mathcal{P}_{\mathbb{RP}^1}={}\tau\nabla^2\). This is the standard real projective structure on \(\mathbb{RP}^1\).

If \(\phi\in \mathrm{ Diff}_+^\infty(\mathbb{RP}^1)\), then one computes that \[S_{{\mathbb{RP}^1}}(\phi)=S_{\mathcal{P}_{\mathbb{RP}^1}}(\phi)= \left(\frac{\phi'''}{\phi'}-\frac{3}{2} \left( \frac{\phi''}{\phi'}\right)^2 \right)\mathrm{d}\tau^2\] where here \(\phi'= \partial_\tau (\tau\circ \phi)\) and likewise for the higher derivatives. This is the classical form of the Schwarzian derivative introduced in §3. Write \(\mathrm{ M\ddot{o}b}(\mathbb{RP}^1)\) for the Möbius group of \(\mathcal{P}_{\mathbb{RP}^1}\) and observe these are the fractional linear transformations. Indeed, if \(\phi\in \mathrm{ M\ddot{o}b}(\mathbb{RP}^1)\), then there is a matrix \[L=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2,\mathbb R)\] so that \[\tau(\phi(p)) = \frac{ a \tau(p)+b}{c \tau(p)+d}.\] This corresponds to the natural action of \(\mathrm{SL}(2,\mathbb R)\) on the space of lines through the origin. Let \[\gamma: \mathrm{SL}(2,\mathbb R) \to \mathrm{ Diff}_+^\infty(\mathbb{RP}^1).\] denote this group homomorphism. Notice that \(\ker \rho=\pm \mathrm{Id}\) and so this map induces an injective homomorphism \[\tilde{\gamma}: \mathrm{PSL}(2,\mathbb R) \to \mathrm{ Diff}_+^\infty(\mathbb{RP}^1)\] whose image is \(\mathrm{ M\ddot{o}b}(\mathbb{RP}^1)\).

Consider the natural map \(T:\mathbb{S}^1 \to \mathbb{RP}^1\) given by sending a point \(p\) to the tangent line to \(\mathbb{S}^1\) through \(p\). Let \({}^\theta\nabla\) be the unique connection on \(\mathbb{S}^1\) so that \(\partial_\theta\) is parallel and let \(\mathcal{P}_{\mathbb{S}^1}={}^\theta\nabla^2\). If \(\phi\in \mathrm{ Diff}_+^\infty(\mathbb{S}^1)\), then one computes that \[S_{\mathbb{S}^1}(\phi)=S_{\mathcal{P}_{\mathbb{S}^1}}(\phi) = \left(\frac{\phi'''}{\phi'}-\frac{3}{2} \left( \frac{\phi''}{\phi'}\right)^2 \right)\mathrm{d}\theta^2\] where here \(\phi'\) has already been defined. Define \(S_{\theta}(\phi)\) so \(S_{\mathbb{S}^1}(\phi)=S_{\theta}(\phi) \mathrm{d}\theta^2.\)

As \(T\circ I=T\), if \(\phi\in\mathrm{ Diff}_+^\infty(\mathbb{S}^1)\) satisfies \(\phi\circ I=I\circ \phi\), then there is a well-defined element \(\tilde{T}(\phi)\in \mathrm{ Diff}_+^\infty(\mathbb{RP}^1)\) so that the following diagram is commutative: \[\begin{CD} \mathbb{S}^1 @>\phi>> \mathbb{S}^1\\ @VV T V @VV T V\\ \mathbb{RP}^1 @>\tilde{T}(\phi)>> \mathbb{RP}^1 \end{CD}\] A straightforward computation shows that, \[S_{\mathbb{S}^1,\mathbb{RP}^1}(T)=S_{\mathcal{P}_{\mathbb{S}^1},\mathcal{P}_{\mathbb{RP}^1}}(T)= 2 \mathrm{d}\theta^2.\] Hence, for a \(\phi\in\mathrm{ Diff}_+^\infty(\mathbb{S}^1)\) which satisfies \(\phi\circ I=I\circ \phi\) the co-cycle relation for the Schwarzian implies \[\begin{aligned} 0&=S_{\mathbb{S}^1, \mathbb{RP}^1}(T\circ \phi)-S_{\mathbb{S}^1, \mathbb{RP}^1}(\tilde{T}(\phi)\circ T) \\ &= 2 \phi^* \mathrm{d}\theta^2 +S_{\mathbb{S}^1}(\phi)-T^* S_{\mathbb{RP}^1}(\tilde{T}(\phi))+2 \mathrm{d}\theta^2\\ &=S_{\mathbb{S}^1}(\phi)+2 (\phi')^2 \mathrm{d}\theta^2 +2 \mathrm{d}\theta^2 -T^* S_{\mathbb{RP}^1}(\tilde{T}(\phi))\end{aligned}\] That is, \[S_{{\mathbb{S}^1}} (\phi) +2(\phi' )^2 \mathrm{d}\theta^2 -2 \mathrm{d}\theta^2=T^* S_{\mathbb{RP}^1} (\tilde{T}(\phi)).\] One verifies from their definitions that \(\tilde{T}(\mathrm{ M\ddot{o}b}(\mathbb{S}^1))=\mathrm{ M\ddot{o}b}(\mathbb{RP}^1)\) and which gives (3.2). Finally, we note the following commutative diagram \[\begin{CD} \mathrm{SL}(2,\mathbb{R}) @>\Gamma>> \mathrm{Mob}(\mathbb{S}^1)\\ @VV \pi V @VV \tilde{T} V\\ \mathrm{PSL}(2,\mathbb{R}) @>\tilde{\gamma}>> \mathrm{Mob}(\mathbb{RP}^1) \end{CD}\] where \(\pi\) is the natural projection.