8 The path geometry defined by a thermostat
A thermostat naturally defines a path geometry and in this final section we show that the path geometry associated to the thermostat coming from a holomorphic differential \(A\) of degree \(m\geqslant 2\) is flat if and only if \(A\) vanishes identically or \(m=3\). The former case corresponds to the paths being the geodesics of a hyperbolic metric and the latter case to the paths being the geodesics of a convex projective structure. We first recall some elementary facts about path geometries while referring the reader to [7] for further details.
An (oriented) path geometry on an oriented surface \(M\) is given by an oriented line bundle \(L\) on the projective circle bundle \(\mathbb{S}M:=\left(TM\setminus\{0\}\right)/\mathbb{R}^+\) having the property that \(L\) together with the vertical bundle of the projection map \(\nu : \mathbb{S}M\to M\) spans the contact distribution of \(\mathbb{S}M\). The paths of \(L\) are the projections of its integral curves to \(M\). Note that the orientation of \(L\) naturally equips its paths with an orientation.
Taking \(M\) to be the oriented \(2\)-sphere \(S^2\), we obtain a canonical path geometry \(L_0\) whose paths are the great circles. In this case \(\mathbb{S}S^2\simeq \mathrm{SO}(3)\) and \(L_0\) is the line bundle defined by \(\omega_2=\psi=0\), where we write the Maurer–Cartan form \(\omega_{\mathrm{SO}(3)}\) of \(\mathrm{SO}(3)\) as \[\omega_{\mathrm{SO}(3)}=\begin{pmatrix} 0 & -\omega_1 & -\omega_2 \\ \omega_1 & 0 & -\psi \\ \omega_2 & \psi & 0\end{pmatrix}\] for left-invariant \(1\)-forms \(\omega_1,\omega_2,\psi\) on \(\mathrm{SO}(3)\). Moreover, we orient \(S^2\) such that an orientation compatible volume form pulls back to \(\mathrm{SO}(3)\) to become a positive multiple of \(\omega_1\wedge\omega_2\) and orient \(L_0\) in such a way that \(\omega_1\) is positive on positive vectors of \(L_0\).
A path geometry \(L\) on \(M\) is called flat, if for every point \(p \in M\), there exists a neighbourhood \(U_p\) and an orientation preserving diffeomorphism \(f : U_p \to V\) onto some open subset \(V\subset S^2\), which maps the positively oriented paths contained in \(U_p\) onto positively oriented great circles.
Let now \(F=X+\lambda V\) be a thermostat on the unit tangent bundle \(SM\) of a oriented Riemannian \(2\)-manifold \((M,g)\). We henceforth identify \(SM \simeq \mathbb{S}M\) in the obvious way. In doing so, we obtain a path geometry by defining \(L:=\mathbb{R}F\) and by declaring vectors in \(L\) to be positive if they are positive multiples of \(F\).
Clearly, if a path geometry is flat, then it must have the property that its paths agree with the geodesics of some projective structure. In [32] it is shown that the path geometry defined by a thermostat \(X+\lambda V\) shares its paths with the geodesics of some projective structure if and only if \[\tag{8.1} 0=\frac{3}{2}\lambda+\frac{5}{3}VV\lambda+\frac{1}{6}VVVV\lambda.\] Using this fact we immediately obtain:
Let \((g,A)\) be a pair satisfying the coupled vortex equations \(\bar{\partial}A=0\) and \(K_g=-1+(m-1)|A|^{2}_{g}\). Then the path geometry defined by the thermostat associated to \((g,A)\) is flat if and only if \(m=3\) or \(A\) vanishes identically.
Proof. Suppose the path geometry associated to \((g,A)\) is flat. Recall that for our choice \(\lambda=a\) we have \(VVa=-m^2 a\), hence (8.1) gives \[0=\left(\frac{1}{6}m^4-\frac{5}{3}m^2+\frac{3}{2}\right)a=\frac{1}{6}(m-1)(m+1)(m-3)(m+3)a.\] Consequently, \(a\) and hence \(A\) must vanish identically or \(m=3\).
Conversely, assume \(A\) is a cubic differential satisfying \(\overline{\partial} A=0\) and \(K_g=-1+2|A|^2_g\). The path geometry associated to \((g,A)\) defines a properly convex projective structure on the oriented surface \(M\). An oriented properly convex projective surface is an example of a surface carrying a \((G,X)\)-structure where \(X=\mathbb{S}^2\) is the oriented projective \(2\)-sphere and \(G=\mathrm{SL}(3,\mathbb{R}\)) its group of projective transformations, cf. [21]. In particular, it follows that the path geometry associated to \((g,A)\) is flat.