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 $m ⩾ 2$ 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 $S M := (T M ∖ {0}) / R^{+}$ having the property that $L$ together with the vertical bundle of the projection map $ν : S M \to M$ spans the contact distribution of $S M$ . 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.

Example 8.1

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 $S S^{2} ≃ SO (3)$ and $L_{0}$ is the line bundle defined by $ω_{2} = ψ = 0$ , where we write the Maurer–Cartan form $ω_{SO (3)}$ of $SO (3)$ as $ω_{SO (3)} = (\begin{matrix} 0 & - ω_{1} & - ω_{2} \\ ω_{1} & 0 & - ψ \\ ω_{2} & ψ & 0 \end{matrix})$ for left-invariant $1$ -forms $ω_{1}, ω_{2}, ψ$ on $SO (3)$ . Moreover, we orient $S^{2}$ such that an orientation compatible volume form pulls back to $SO (3)$ to become a positive multiple of $ω_{1} \land ω_{2}$ and orient $L_{0}$ in such a way that $ω_{1}$ is positive on positive vectors of $L_{0}$ .

Definition 8.2

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 $F = X + λ V$ be a thermostat on the unit tangent bundle $S M$ of a oriented Riemannian $2$ -manifold $(M, g)$ . We henceforth identify $S M ≃ S M$ in the obvious way. In doing so, we obtain a path geometry by defining $L := R F$ 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 $X + λ V$ shares its paths with the geodesics of some projective structure if and only if $\begin{matrix} (8.1) & 0 = \frac{3}{2} λ + \frac{5}{3} V V λ + \frac{1}{6} V V V V λ . \end{matrix}$ Using this fact we immediately obtain:

Theorem 8.3

Let $(g, A)$ be a pair satisfying the coupled vortex equations $\bar{\partial} A = 0$ and $K_{g} = - 1 + (m - 1) | A |_{g}^{2}$ . 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 $λ = a$ we have $V V a = - m^{2} a$ , hence (8.1) gives $0 = (\frac{1}{6} m^{4} - \frac{5}{3} m^{2} + \frac{3}{2}) a = \frac{1}{6} (m - 1) (m + 1) (m - 3) (m + 3) a .$ Consequently, $a$ and hence $A$ must vanish identically or $m = 3$ .

Conversely, assume $A$ is a cubic differential satisfying $\overset{―}{\partial} A = 0$ and $K_{g} = - 1 + 2 | A |_{g}^{2}$ . 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 $X = S^{2}$ is the oriented projective $2$ -sphere and $G = SL (3, R$ ) its group of projective transformations, cf. [21]. In particular, it follows that the path geometry associated to $(g, A)$ is flat.