2 Preliminaries
2.1 The unit tangent bundle
Let \((M,g)\) be an oriented Riemannian \(2\)-manifold and let \(\pi : SM \to M\) denote its unit tangent bundle. Recall that \(SM\) is equipped with a coframing consisting of three linearly independent \(1\)-forms \((\underline{\omega_1},\underline{\omega_2},\underline{\psi})\). The \(1\)-forms \((\underline{\omega_1},\underline{\omega_2})\) span the \(1\)-forms on \(SM\) that are semibasic for the basepoint projection \(\pi\), that is, the forms that vanish when evaluated on vertical vector fields. Explicitly, we have for all \((x,v) \in SM\) and \(\xi \in T_{(x,v)}SM\) \[\underline{\omega_1}(\xi)=g(d\pi(\xi),v)\quad\text{and}\quad \underline{\omega_2}(\xi)=g(d\pi(\xi),Jv),\] where \(J : TM \to TM\) denotes rotation by \(\pi/2\) in counter-clockwise direction with respect to the fixed orientation. The third \(1\)-form \(\underline{\psi}\) is the Levi-Civita connection form of \(g\) so that we have the structure equations \[\mathrm{d}\underline{\omega_1}=-\underline{\omega_2}\wedge\underline{\psi},\qquad \mathrm{d}\underline{\omega_2}=-\underline{\psi}\wedge\underline{\omega_1},\qquad \mathrm{d}\underline{\psi}=-K_g\underline{\omega_1}\wedge\underline{\omega_2},\] where \(K_g\) denotes the (pullback to \(SM\) of the) Gauss curvature of \(g\). Denoting by \((\underline{X},\underline{H},\underline{V})\) the vector fields dual to \((\underline{\omega_1},\underline{\omega_2},\underline{\psi})\), the structure equations imply the commutator relations \[\tag{2.1} [\underline{V},\underline{X}]=\underline{H},\qquad [\underline{V},\underline{H}]=-\underline{X}, \qquad [\underline{X},\underline{H}]=K_g\underline{V}.\] The vector field \(\underline{X}\) is the geodesic vector field of \((M,g)\) and \(\underline{V}\) is the generator of the \(\mathrm{SO}(2)\) right action on \(SM\) which we denote by \(R_{e^{\mathrm{i}\vartheta}}\) for \(e^{\mathrm{i}\vartheta} \in \mathrm{SO}(2)\).
Note that a complex-valued \(1\)-form on \(M\) that is a \((1,\! 0)\)-form with respect to the Riemann surface structure defined by \(J\) pulls back to \(SM\) to become a complex multiple of the form \(\underline{\omega}:=\underline{\omega_1}+\mathrm{i}\underline{\omega_2}\). The form \(\underline{\omega}\) satisfies the equivariance property \((R_{e^{\mathrm{i}\vartheta}})^*\underline{\omega}=e^{-\mathrm{i}\vartheta}\underline{\omega}\) for all \(e^{\mathrm{i}\vartheta} \in \mathrm{SO}(2)\) and hence a section \(\beta\) of the canonical bundle \(K\) of \(M\) is represented by a complex-valued function \(\boldsymbol\beta\) on \(SM\) satisfying the equivariance property \((R_{e^{\mathrm{i}\vartheta}})^*\boldsymbol\beta=e^{\mathrm{i}\vartheta}\boldsymbol\beta\). To recover the associated \((1,\! 0)\)-form on \(M\), we observe that \(\boldsymbol\beta\underline{\omega}\) is semi-basic and invariant under the \(\mathrm{SO}(2)\)-right action, hence the pullback of a unique \((1,\! 0)\)-form on \(M\), which is \(\beta\).
We write \(Y(f)\) for the (Lie-)derivative of a smooth real – or complex-valued function \(f\) in the direction of a vector field \(Y\). Whenever no confusion is possible about the argument of the linear differential operator \(Y\), we will simply write \(Yf\) instead of \(Y(f)\).
2.2 Roots of the unit tangent bundle
Let \(n \in \mathbb{N}\) and \(\pi_{n} : SM^{1/n} \to M\) be a principal right \(\mathrm{SO}(2)\)-bundle whose right action we denote by \(R_{e^{\mathrm{i}\vartheta}}\) as well. Let \(\pi : SM \to M\) denote the unit tangent bundle of the oriented Riemannian \(2\)-manifold \((M,g)\) and \((\underline{\omega_1},\underline{\omega_2},\underline{\psi})\) its coframing. We call \(\pi_{n} : SM^{1/n} \to M\) an \(n\)-th root of \(SM\) if there exists an \(n\)-fold covering map \(\rho : SM^{1/n} \to SM\) so that \(\pi_{n}=\pi \circ \rho\) and so that \[\rho\circ R_{e^{\mathrm{i}\vartheta}}=R_{e^{\mathrm{i}n\vartheta}}\circ \rho\] for all \(e^{\mathrm{i}\vartheta} \in \mathrm{SO}(2)\). We refer the reader to [15] for background about \(n\)-th roots of \(SM\). We write \(\omega_i=\rho^*\underline{\omega_i}\) and \(\psi=\rho^*\underline{\psi}\) and let \((X,H,\mathbb{V})\) denote the framing dual to \((\omega_1,\omega_2,\psi)\) on \(SM^{1/n}\). The structure equations imply the usual commutator relations \[\tag{2.2} [\mathbb{V},X]=H,\qquad [\mathbb{V},H]=-X, \qquad [X,H]=K_g\mathbb{V}.\] Recall that a section \(\beta\) of the canonical bundle \(K\) of \((M,g)\) is represented by a complex-valued function \(\boldsymbol\beta\) on \(SM\) satisfying the equivariance property \((R_{e^{\mathrm{i}\vartheta}})^* \boldsymbol\beta=e^{\mathrm{i}\vartheta} \boldsymbol\beta\). Writing \(\tilde{ \boldsymbol\beta}:=\boldsymbol\beta\circ \rho\), the function \(\tilde{\boldsymbol\beta}\) satisfies \((R_{e^{\mathrm{i}\vartheta}})^*\tilde{\boldsymbol\beta}=e^{\mathrm{i}n\vartheta}\tilde{\boldsymbol\beta}\) and hence we obtain a \(n\)-th root \(K^{1/n}\) of \(K\) whose sections are represented by complex-valued functions \(\boldsymbol{B}\) on \(SM^{1/n}\) satisfying \((R_{e^{\mathrm{i}\vartheta}})^* \boldsymbol{B}=e^{\mathrm{i}\vartheta} \boldsymbol{B}\) for all \(e^{\mathrm{i}\vartheta} \in \mathrm{SO}(2)\). Likewise, for each \(m \in \mathbb{Z}\), the smooth sections of \(K^{m/n}\) are represented by smooth complex-valued functions \(\boldsymbol{B}\) on \(SM^{1/n}\) satisfying \[\tag{2.3} (R_{e^{\mathrm{i}\vartheta}})^* \boldsymbol{B}=e^{\mathrm{i}m\vartheta} \boldsymbol{B}\] for all \(e^{\mathrm{i}\vartheta} \in \mathrm{SO}(2)\). In particular, for each \(m \in \mathbb{Z}\) we obtain a Hermitian bundle metric \(\mathcal{h}_0\) on \(K^{m/n}\) defined by \[(\boldsymbol{B}_1,\boldsymbol{B}_2) \mapsto \boldsymbol{B}_1\overline{\boldsymbol{B}_2},\] where \(\boldsymbol{B}_1,\boldsymbol{B}_2\) represent sections of \(K^{m/n}\).
Furthermore, observe that by definition, \(\mathbb{V}\) is only \((1/n)\)-th of the generator \(V\) of the \(\mathrm{SO}(2)\)-action on \(SM^{1/n}\). As a consequence, the infinitesimal version of (2.3) becomes \[\tag{2.4} \mathbb{V}\boldsymbol{B}=\frac{1}{n}V\boldsymbol{B}=\mathrm{i}\left(\frac{m}{n}\right)\boldsymbol{B}\] and hence the map \[\boldsymbol{B}\mapsto \mathrm{d}\boldsymbol{B}-\mathrm{i}\left(\frac{m}{n}\right)\psi \boldsymbol{B}\] equips \(K^{m/n}\) with a connection \(\nabla\) whose connection form is \(-\mathrm{i}(m/n)\psi\). The \((0,\! 1)\)-part \(\nabla^{\prime\prime}\) of \(\nabla\) equips \(K^{m/n}\) with a holomorphic line bundle structure \(\overline{\partial}_{K^{m/n}}\), so that \(\nabla\) is the Chern connection of the Hermitian holomorphic line bundle \((K^{m/n},\overline{\partial}_{K^{m/n}},\mathcal{h}_0)\).
Finally, note that applying \(\mathbb{V}\) again to (2.4) shows that we may write \(\boldsymbol{B}=\frac{n\mathbb{V}b}{m}+\mathrm{i}b\) for a unique real-valued function \(b\) on \(SM^{1/n}\) satisfying \(\mathbb{V}\mathbb{V}b=-\left(\frac{m}{n}\right)^2 b\). Conversely, if a smooth real-valued function \(b\) on \(SM^{1/n}\) satisfies \(\mathbb{V}\mathbb{V}b=-(\frac{m}{n})^2 b\), then \(\boldsymbol{B}:=\frac{n\mathbb{V}b}{m}+\mathrm{i}b\) represents a smooth section \(B\) of \(K^{m/n}\).
2.3 Thermostats
Let \(N\) be a smooth \(3\)-manifold equipped with three smooth vector fields \((X,H,V)\) that are linearly independent at each point of \(N\). Following [8] we define:
We say \(N\) carries a generalised Riemannian structure if \((X,H,V)\) satisfy the commutator relations \[\tag{2.5} [V,X]=H,\qquad [V,H]=-X, \qquad [X,H]=K_g V,\] for some smooth function \(K_g\) on \(N\).
Let \((M,g)\) be an oriented Riemannian \(2\)-manifold and \(\pi_n : SM^{1/n} \to M\) an \(n\)-th root of its unit tangent bundle \(\pi : SM \to M\). Then \((X,H,\mathbb{V})\) defined as in 2.2 equip \(N=SM^{1/n}\) with a generalised Riemannian structure.
Suppose \(N\) carries a generalised Riemannian structure \((X,H,V)\) with dual \(1\)-forms \((\omega_1,\omega_2,\psi)\).
A (generalised) thermostat on \(N\) is a flow \(\phi\) generated by a vector field of the form \(F:=X+\lambda V\), where \(\lambda \in C^{\infty}(N)\).