3 Projective thermostats
In this section we show how to associate the triple \((g,A,\theta)\) to a given projective structure \(\mathfrak{p}\). As mentioned in the introduction, the existence of such a triple is a consequence of some elementary facts about \(\mathrm{SO}(2)\)-representation theory and a description of projective structures as sections of a certain affine bundle over \(M\) (see [17] for a construction of \((g,A,\theta)\) in that spirit), here instead we obtain the triple as a by-product of a characterisation of projective thermostats.
A (generalised) thermostat is a flow \(\phi\) on \(SM\) which is generated by a vector field of the form \(F=X+\lambda V\), where \(\lambda\) is a smooth real-valued function on \(SM\). In this article we are mainly interested in the case where the generalised thermostat is projective. By this we mean that there exists a torsion-free connection \(\nabla\) on \(TM\) having the property that for every \(\phi\)-orbit \(\gamma : I \to SM\), there exists a reparametrisation \(\varphi : I^{\prime} \to I\) so that \(\pi\circ\gamma\circ\varphi : I^{\prime} \to M\) is a geodesic of \(\nabla\).
Phrased more loosely, the orbit projections to \(M\) agree with the geodesics of a projective structure \(\mathfrak{p}\) on \(M\). By a classical result of Cartan, Eisenhart, Weyl (see for instance [23] for a modern reference), two torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) are projectively equivalent if and only if there exists a \(1\)-form \(\alpha\) on \(M\) so that \[\nabla^{\prime}-\nabla=\mathrm{Sym}(\alpha).\]
3.1 A characterisation of projective thermostats
It turns out that projective thermostats admit a simple characterisation in terms of the vertical Fourier decomposition (2.7) of \(\lambda\). Towards this end we first show:
Let \(\nabla\) be a torsion-free connection on the tangent bundle \(TM\) and \(\varphi=(\varphi^i_j) \in \Omega^1(SM,\mathfrak{gl}(2,\mathbb{R}))\) its connection form. Then, up to reparametrisation, the leaves of the foliation \(\mathcal{F}\) defined by \(\varphi^2_1=\omega_2=0\) project to \(M\) to become the geodesics of \(\nabla\). Conversely, every geodesic of \(\nabla\), parametrised with respect to \(g\)-arc length, lifts to become a leaf of \(\mathcal{F}\).
Proof. Recall that the set of torsion-free connections on \(TM\) is an affine space modelled on the sections of \(S^2(T^*M)\otimes TM\). It follows that there exists a \(1\)-form \(\tilde{B}\) on \(M\) with values in the endomorphisms of \(TM\) so that \(\nabla={}^g\nabla+\tilde{B}\). As we have seen, the connection form of the Levi-Civita connection of \(g\) on \(TM\) is \[\kappa=\begin{pmatrix} 0 & -\psi \\ \psi & 0 \end{pmatrix}.\] Hence there exist unique real-valued function \(b^i_{jk}=b^i_{kj}\) on \(SM\) so that \[\varphi=\begin{pmatrix} 0 & -\psi \\ \psi & 0 \end{pmatrix}+\begin{pmatrix} b^1_{11}\omega_1+b^1_{12}\omega_2 & b^1_{21}\omega_1+b^1_{22}\omega_2 \\ b^2_{11}\omega_1+b^2_{12}\omega_2 & b^2_{21}\omega_1+b^2_{22}\omega_2\end{pmatrix}.\] Explicitly, \(b^i_{jk}(v)=g(\tilde{B}(e_j)e_k,e_i)\), where we write \(e_1=v\) and \(e_2=J v\) for \(v \in SM\).
Let \(\delta : I \to SM\) be a leaf of \(\mathcal{F}\), so that \(\delta^*\omega_2=0\). Writing \(\gamma:=\pi \circ \delta\) and evaluating \(\delta^*\omega_2\) on the standard vector field \(\partial_t\) of \(\mathbb{R}\), we obtain \[0=\partial_t \hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\delta^*\omega_2=g\left(d(\pi \circ \delta)(\partial_t),J\delta(t)\right)=g(\dot{\gamma}(t),J\delta(t)),\] so that \(\delta=f\dot{\gamma}\) for some unique \(f \in C^{\infty}(I)\). Hence without losing generality, we may assume that the leaves of \(\mathcal{F}\) are of the form \(\dot{\gamma}\) for some smooth curve \(\gamma : I \to M\) having unit length velocity vector with respect to \(g\).
By construction of \(\psi\), see (2.6), the pullback \(1\)-form \(\dot{\gamma}^*\psi\) evaluated on \(\partial_t\) gives the function \(g({}^g\nabla_{\dot{\gamma}}\dot{\gamma},J\dot{\gamma})\), hence \(\dot{\gamma}^*\varphi^2_1=0\) if and only if \[0=g\left({}^g\nabla_{\dot{\gamma}}\dot{\gamma},J\dot{\gamma}\right)+b^2_{11}(\dot{\gamma})=g\left({}^g\nabla_{\dot{\gamma}}\dot{\gamma}+\tilde{B}(\dot{\gamma})\dot{\gamma},J\dot{\gamma}\right).\] It follows that there exists a function \(f \in C^{\infty}(I)\) so that \[{}^g\nabla_{\dot{\gamma}}\dot{\gamma}+\tilde{B}(\dot{\gamma})\dot{\gamma}=\nabla_{\dot{\gamma}}\dot{\gamma}=f\dot{\gamma}.\] By a standard lemma in projective differential geometry [23] a smooth immersed curve \(\gamma : I \to M\) can be reparametrised to become a geodesic of the torsion-free connection \(\nabla\) on \(TM\) if and only if there exists a smooth function \(f : I \to \mathbb{R}\) so that \(\nabla_{\dot{\gamma}}\dot{\gamma}=f\dot{\gamma}\). The claim follows by applying this lemma.
Suppose the thermostat \(F=X+\lambda V\) is projective, then \[0=\frac{3}{2}\lambda+\frac{5}{3}VV\lambda+\frac{1}{6}VVVV\lambda.\]
Proof. Using notation as in the proof of Lemma 3.1, we must have \(F\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\varphi^2_1=0\) and \(F\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\omega_2=0\). The latter conditions is trivially satisfied, but the former gives \[F\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\varphi^2_1=\left(X+\lambda V\right)\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\left(\psi+b^2_{11}\omega_1+b^2_{12}\omega_2\right)=\lambda+b^2_{11}=0,\] so that \(\lambda=-b^2_{11}\). Since the functions \(b^i_{jk}\) represent a section of \(S^2(T^*M)\otimes TM\), they satisfy the structure equations \[db^i_{jk}=b^i_{jl}\kappa^l_k+b^i_{lk}\kappa^l_j-b^l_{jk}\kappa^i_l, \quad \text{mod}\quad \omega_i.\] In particular, from this we compute \[Vb^2_{11}=V\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}db^2_{11}=V\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\left(2b^2_{12}-b^1_{11}\right)\psi=2b^2_{12}-b^1_{11}.\] Applying \(V\) again we obtain \[VVb^2_{11}=2b^2_{22}-3b^2_{11}-4b^1_{12}\] and likewise \[VVVVb^2_{11}=40b^1_{12}+21b^2_{11}-20b^2_{22},\] so that the claim follows from an elementary calculation.
For \(\lambda \in C^{\infty}(SM)\) the following statements are equivalent:
\(0=\frac{3}{2}\lambda+\frac{5}{3}VV\lambda+\frac{1}{6}VVVV\lambda\);
\(\lambda \in \mathcal{H}_{-1}\oplus \mathcal{H}_{1}\oplus \mathcal{H}_{-3}\oplus \mathcal{H}_{3}\).
Proof. Let \(\lambda \in \mathcal{H}_{-3}\oplus \mathcal{H}_{-1}\oplus \mathcal{H}_{1}\oplus \mathcal{H}_{3}\) so that we may write \(\lambda=\lambda_{-3}+\lambda_{-1}+\lambda_1+\lambda_3\) with \(\lambda_m \in \mathcal{H}_m\). Since \(\lambda\) is real-valued we have \(\lambda_{-1}=\overline{\lambda_1}\) and \(\lambda_{-3}=\overline{\lambda_3}\). Hence setting \(\nu_1=\lambda_{-1}+\lambda_1\) and \(\nu_3=\lambda_{-3}+\lambda_3\), we obtain \(VV\nu_1=-\nu_1\) and \(VV\nu_3=-9\nu_3\) so that \[\frac{3}{2}\lambda+\frac{5}{3}VV\lambda+\frac{1}{6}VVVV\lambda=\frac{3}{2}(\nu_3+\nu_1)+\frac{5}{3}(-9\nu_3-\nu_1)+\frac{1}{6}(81\nu_3+\nu_1)=0.\] Conversely, suppose \(\lambda \in C^{\infty}(SM)\) satisfies \(0=\frac{3}{2}\lambda+\frac{5}{3}VV\lambda+\frac{1}{6}VVVV\lambda\) and write \(\lambda=\sum_{m}\lambda_m\) with \(\lambda_m \in \mathcal{H}_m\). Hence we obtain \[0=\frac{3}{2}\lambda+\frac{5}{3}VV\lambda+\frac{1}{6}VVVV\lambda =\sum_{m}\left(\frac{3}{2}-\frac{5}{3}m^2+\frac{1}{6}m^4\right)\lambda_m\] so that \(\lambda_m=0\) unless \[0=\frac{3}{2}-\frac{5}{3}m^2+\frac{1}{6}m^4=\frac{1}{6}(m-3)(m-1)(m+1)(m+3).\] The claim follows.
Finally, we obtain:
A thermostat \(F=X+\lambda V\) is projective if and only if \(\lambda \in \mathcal{H}_{-1}\oplus \mathcal{H}_{1}\oplus \mathcal{H}_{-3}\oplus \mathcal{H}_{3}.\)
Proof. It remains to show that if \(\lambda \in \mathcal{H}_{-1}\oplus \mathcal{H}_{1}\oplus \mathcal{H}_{-3}\oplus \mathcal{H}_{3}\), then there exists a torsion-free connection \(\nabla\) on \(TM\) so that \(F \hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\varphi^2_1\) vanishes identically, where \(\varphi=(\varphi^i_j)\) denotes the connection form of \(\nabla\). We may write \[\lambda=a-V\theta\] where \(a \in C^{\infty}(SM)\) satisfies \(9a+VVa=0\) and \(\theta\) is a smooth \(1\)-form on \(M\), thought of as a real-valued function on \(SM\). Since \(9a+VVa=0\), there exists a unique cubic differential \(A\) on \(M\) so that \(\pi^*A=(Va/3+ia)\omega^3\). Hence simple computations show that \[\tag{3.1} \begin{aligned} a(v)&=\operatorname{Re}A(Jv,Jv,Jv)=-\operatorname{Re}A(Jv,v,v)\\ \frac{1}{3}Va(v)&=\operatorname{Re}A(v,v,v)=-\operatorname{Re}A(v,Jv,Jv) \end{aligned}\] for all \(v \in SM\). Let \(B\) be the unique \(1\)-form on \(M\) with values in the endomorphisms of \(TM\) satisfying \[\tag{3.2} g(B(v_1)v_2,v_3)=\mathrm{Re}\, A(v_1,v_2,v_3)\] for all tangent vectors \(v_1,v_2,v_3 \in TM\). On \(TM\) consider the torsion-free connection \(\nabla=\mathrm{D}+B\), where \(\mathrm{D}\) is the Weyl connection \[\mathrm{D}={}^g\nabla+g\otimes \theta^{\sharp}-\mathrm{Sym}(\theta).\] Using (2.8) and (3.1), we compute that the connection form of \(\nabla\) is \[\tag{3.3}\begin{aligned} \varphi=\begin{pmatrix} -\theta\omega_1-V(\theta)\omega_2 & -V(\theta)\omega_1+\theta\omega_2 -\psi \\ \psi+V(\theta)\omega_1-\theta\omega_2 & -\theta\omega_1-V(\theta)\omega_2\end{pmatrix}\\ +\begin{pmatrix} V(a)/3\omega_1-a\omega_2 & -a\omega_1-V(a)/3\omega_2 \\ -a\omega_1-V(a)/3\omega_2 & -V(a)/3\omega_1+a\omega_2\end{pmatrix}.\end{aligned}\] In particular, we have \[\varphi^2_1=\psi+\left(V(\theta)-a\right)\omega_1-\left(\theta+V(a)/3\right)\omega_2,\] so that \(F\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\varphi^2_1=0\).
3.2 The effect of a conformal change
Summarising the previous subsection, we have seen that if \(\nabla\) is a torsion-free connection on \(TM\) and we fix a Riemannian metric \(g\) on \(M\), then we may write \(\nabla={}^g\nabla+\tilde{B}\) for some endomorphism-valued \(1\)-form \(\tilde{B}\) on \(M\). The thermostat on \(SM\) defined by \(\lambda=-b^2_{11}\) has the property that its orbits project to \(M\) to become the geodesics of \(\nabla\) up to parametrisation. Moreover, we obtain a \(1\)-form \(\theta \in \Omega^1(M)\) as well as a cubic differential \(A \in \Gamma(K^3)\), so that the connection \(\nabla\) shares its geodesics – up to parametrisation – with the projections to \(M\) of the orbits of the projective thermostat defined by \(\lambda=a-V\theta\), where \(a\) represents the imaginary part of \(A\).
Next we compute how \(\theta\) and \(A\) transform under conformal change of the metric. As a consequence, we obtain:
Let \(\nabla\) be a torsion-free connection on \(TM\). Then the choice of a conformal structure \([g]\) on \(M\) determines a unique Weyl connection \(\mathrm{D}\) for \([g]\) and a unique section \(\Phi\) of \(K^2\otimes\overline{K^*}\) so that \(\mathrm{D}+\operatorname{Re}\Phi\) is projectively equivalent to \(\nabla\).
Proof. Let \(g \mapsto \hat{g}=\mathrm{e}^{2u}g\) be a conformal change of the metric, where \(u \in C^{\infty}(M)\). For the new metric \(\hat{g}\) there exists a \(1\)-form \(\hat{\theta}\) and a cubic differential \(\hat{A}\) on \(M\) so that \(\mathrm{D}+B\) and \(\hat{\mathrm{D}}+\hat{B}\) are projectively equivalent. Here \(\hat{B}\) denotes the \(1\)-form constructed from \(\hat{A}\) by using the metric \(\hat{g}\). Projective equivalence corresponds to the existence of a \(1\)-form \(\alpha\) on \(M\) so that \[\mathrm{D}+B=\hat{\mathrm{D}}+\hat{B}+\mathrm{Sym}(\alpha)\] Using (2.9) as well as (see [1]) \[\tag{3.4} {}^{\exp(2u)g}\nabla={}^g\nabla-g\otimes{}^g\nabla u+\mathrm{Sym}(du)\] this is equivalent to \[\begin{gathered} {}^g\nabla+g\otimes\theta^{\sharp}-\mathrm{Sym}(\theta)+B={}^g\nabla-g\otimes{}^g\nabla u\\+\mathrm{Sym}(du)+\mathrm{e}^{2u}g\otimes\hat{\theta}^{\hat{\sharp}}-\mathrm{Sym}(\hat{\theta})+\hat{B}+\mathrm{Sym}(\alpha)\end{gathered}\] or \[g\otimes\left(\theta^{\sharp}+{}^g\nabla u-\hat{\theta}^{\sharp}\right)+B-\hat{B}=\mathrm{Sym}\left(\beta\right),\] where \(\beta=\alpha+\theta+du-\hat{\theta}\). Evaluating this equation on the pair \((v,Jv)\) with \(v\) a unit tangent vector with respect to \(g\) gives \[B(v)Jv-\hat{B}(v)Jv=\mathrm{Sym}\left(\beta\right)(v,Jv).\] Computing the inner product with the tangent vector \(v\) yields \[\mathrm{Re}\,A(v,Jv,v)-\mathrm{e}^{-2u}\mathrm{Re}\,\hat{A}(v,Jv,v)=\beta(Jv).\] Thought of as an identity for functions on \(SM\), the left hand side lies in \(\mathcal{H}_{-3}\oplus\mathcal{H}_3\) whereas the right hand side lies in \(\mathcal{H}_{-1}\oplus \mathcal{H}_1\) and hence they can only be equal if both sides vanish identically. Consequently, it follows that \(\beta=0\) and that \[\hat{A}=\mathrm{e}^{2u}A.\] Therefore, \(B=\hat{B}\) and \[\tag{3.5} \hat{\theta}=\theta+du\] so that \(\alpha=0\) as well as \(\mathrm{D}=\hat{\mathrm{D}}\).
In particular, we see that both \(\mathrm{D}\) and \(B\) do only depend on the conformal equivalence class of \(g\). We may define a section \(\Phi\) of \(K^2\otimes \overline{K^*}\) by \(\Phi d\sigma=A\), where \(d\sigma\) denotes the area form of \(g\). Comparing with (3.2), we see that \(B\) is the real part of \(\Phi\).