3 \(\mathrm{GL}(2)\)-structures
Let \(x,y\) denote the standard linear coordinates on \(\mathbb{R}^2\) and let \(\mathbb{R}[x,y]\) denote the polynomial ring with real coefficients generated by \(x\) and \(y\). We let \(\mathrm{GL}(2,\mathbb{R})\) act from the left on \(\mathbb{R}[x,y]\) via the usual linear action on \(x,y\). We denote by \(\mathcal{V}_d\) the subspace consisting of homogeneous polynomials in degree \(d\geqslant 0\) and by \(G_d\subset \mathrm{GL}(\mathcal{V}_d)\) the image subgroup of the \(\mathrm{GL}(2,\mathbb{R})\) action on \(\mathcal{V}_3\). The vector space \(\mathcal{V}_3\) carries a two-dimensional cone \(\tilde{\mathcal{C}}\) of distinguished polynomials, consisting of the perfect cubes, i.e., those that are of the form \((ax+by)^3\) for \(ax+by\in \mathcal{V}_1\). The reader may easily check that \(G_3\) is characterised as the subgroup of \(\mathrm{GL}(\mathcal{V}_3)\) that preserves \(\tilde{\mathcal{C}}\). The projectivisation of \(\tilde{\mathcal{C}}\) gives an algebraic curve \(\mathcal{C}\) of degree \(3\) in \(\mathbb{P}(\mathcal{V}_3)\), which is linearly equivalent to the twisted cubic curve, i.e., the curve in \(\mathbb{RP}^3\) defined by the zero locus of the three homogeneous polynomials \[P_0=XZ-Y^2, \quad P_1=YW-Z^2, \quad P_2=XW-YZ,\] where \([X\!:\!Y\!:\!Z\!:\!W]\) are the standard homogeneous coordinates on \(\mathbb{RP}^3\). The vector space \(\mathcal{V}_3\) carries another algebraic variety in its projectivisation besides the twisted cubic curve. Indeed, the polynomials having vanishing discriminant define a \(G_3\)-invariant quartic cone \(\tilde{\mathcal{Q}}\) whose projectivisation \(\mathcal{Q}\) defines a quartic hypersurface in \(\mathbb{P}(\mathcal{V}_3)\). Furthermore, the singular locus of \(\mathcal{Q}\) is the twisted cubic curve \(\mathcal{C}\) and the tangential variety of \(\mathcal{C}\) is \(\mathcal{Q}\).
Let \(M\) be a \(4\)-manifold and let \(\upsilon \colon FM \to M\) denote its coframe bundle modelled on \(\mathcal{V}_3\). A \(\mathrm{GL}(2)\)-structure on \(M\) is a reduction \(\pi \colon B \to M\) of \(FM\) with structure group \(G_3\simeq \mathrm{GL}(2,\mathbb{R})\). By definition, a \(\mathrm{GL}(2)\)-structure identifies each tangent space of \(M\) with \(\mathcal{V}_3\) up to the action by \(\mathrm{GL}(2,\mathbb{R}\)). Consequently, each projectivised tangent space \(\mathbb{P}(T_pM)\) of \(M\) carries an algebraic curve \(\mathcal{C}_p\), which is linearly equivalent to the twisted cubic curve. Conversely, if \(\mathcal{C}\subset \mathbb{P}(TM)\) is a smooth subbundle having the property that each fibre \(\mathcal{C}_p\) is linearly equivalent to the twisted cubic curve, then one obtains a unique reduction of the coframe bundle of \(M\) whose structure group is \(G_3\).
For what follows it will be convenient to identify \(\mathcal{V}_3\simeq \mathbb{R}^4\) by the isomorphism \(\mathcal{V}_3 \to \mathbb{R}^4\) defined on the basis of monomials as \[x^{(3-i)}y^i \mapsto e_{i+1},\] where \(i=0,1,2,3\) and \(e_i\) denotes the standard basis of \(\mathbb{R}^4\). Note that, under the identification \(T_pM=\mathcal{V}_3\), the cone \(\tilde{\mathcal{C}}\) of a \(\mathrm{GL}(2)\)-structure at \(p\) can be written as \[\tilde{\mathcal{C}}_p=\{s^3e_1+3s^2te_2+3st^2e_3+t^3e_4\ |\ s,t\in\mathbb{R}\}.\] We now have:
All torsion-free \(\mathrm{GL}(2)\)-structures in dimension four are \(H\)-flat, where \(H\subset \mathrm{SL}(4,\mathbb{R})\) is the subgroup consisting of matrices of the form \[\tag{3.1} \begin{pmatrix} 1 & A & B & D \\ 0 & 1 & A & C \\ 0 & 0 & 1 & A \\ 0 & 0 & 0 & 1\end{pmatrix}\] and where \(A,B,C,D\) are arbitrary real numbers.
We note that the group \(H\) is isomorphic to a semidirect product of the continuous three-dimensional Heisenberg group \(H_3(\mathbb{R})\) and the Abelian group \(\mathbb{R}\), that is, \(H\simeq H_3(\mathbb{R})\rtimes \mathbb{R}\). Indeed, \(H_3(\mathbb{R})\) has a faithful (necessarily reducible) four-dimensional representation defined by the Lie group homomorphism \(\varphi \colon H_3(\mathbb{R}) \to \mathrm{SL}(4,\mathbb{R})\) \[\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}\mapsto \begin{pmatrix} 1 & a & \frac{1}{2}a^2+b & \frac{1}{6}a^3+ab-c\\ 0 & 1 & a & \frac{1}{2}a^2 \\ 0 & 0 & 1 & a \\ 0 & 0 & 0 & 1\end{pmatrix}.\] The homomorphism \(\varphi\) embeds \(H_3(\mathbb{R})\) as a normal subgroup of the group \(H\) and we think of \(\mathbb{R}\) as the Abelian subgroup of \(H\) defined by setting \(A=B=D=0\) in (3.1).
In fact, the notion of a \(\mathrm{GL}(2)\)-structure makes sense in all dimensions \(d\geqslant 3\). However, torsion-free \(\mathrm{GL}(2)\)-structures in dimensions exceeding four are \(\{e\}\)-flat [2], that is, flat in the usual sense. We refer the reader to [9, 18] for a comprehensive study of five-dimensional \(\mathrm{GL}(2)\)-structures (with torsion).
Phrased differently, Theorem 3.1 states that locally every torsion-free \(\mathrm{GL}(2)\)-structure in dimension four is obtained from a solution to the first order PDE system (2.2), where \(h\) takes values in the aforementioned group \(H\).
Proof of Theorem 3.1. We shall prove that for a given torsion-free \(\mathrm{GL}(2)\)-structure one can always choose local coordinates such that the cone \(\tilde{\mathcal{C}}\) has the following form \[\tilde{\mathcal{C}}=\{\ s^3V_0+3s^2tV_1+3st^2V_2+t^3V_3|\ s,t\in\mathbb{R}\},\] where the framing \((V_0,V_1,V_2,V_3)\) is \[\tag{3.2} \begin{aligned} &V_0=\partial_0,\qquad V_1=\partial_1+\alpha\partial_0,\qquad V_2=\partial_2+\alpha\partial_1+\beta\partial_0,\\ &V_3=\partial_3+\alpha\partial_2+\gamma\partial_1+\delta\partial_0, \end{aligned}\] for some functions \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Then, the dual coframing is of the form \(h\,\mathrm{d}x\), where \(h\) takes values in \(H\) with \[A=-\alpha,\quad B=-\beta+\alpha^2,\quad C=-\gamma+\alpha^2, \quad D=-\delta+\alpha(\gamma+\beta)-\alpha^3.\]
In order to derive the desired form of \(\tilde{\mathcal{C}}\) we explore a correspondence between the torsion-free \(\mathrm{GL}(2)\)-structures and classes of contact equivalent fourth order ODEs (compare the proof of [4] and a similar correspondence in dimension 3). Indeed, it is proved in [2] that any torsion-free \(\mathrm{GL}(2)\)-structure is defined by a fourth order ODE of the form \[\tag{3.3} x^{(4)}=F(y,x,x',x'',x'''),\] where the function \(F=F(y,x_0,x_1,x_2,x_3)\) satisfies a system of non-linear equations that we will refer to as the Bryant–Wünschmann condition. (Similar conditions in higher dimensions are known as the generalized Wünschmann conditions, because they generalize the classical 3-dimensional case, c.f. [6, 17].)
Above, \((y,x_0,x_1,x_2,x_3)\) denote the standard coordinates on the space \(J^3(\mathbb{R},\mathbb{R})\) of 3-jets of functions \(\mathbb{R}\to\mathbb{R}\) and the Bryant–Wünschmann condition is invariant with respect to the group of contact transformations of the coordinates. The \(\mathrm{GL}(2)\)-structure corresponding to equation (3.3) is defined on the solution space of (3.3), i.e., on the quotient space \(J^3(\mathbb{R},\mathbb{R})/X_F\), where \(X_F=\partial_y+x_1\partial_0+x_2\partial_1+x_3\partial_2+F\partial_3\) is the total derivative. In order to define the structure, we first consider the following field of cones on \(J^3(\mathbb{R},\mathbb{R})\) as in [12] \[\hat{\mathcal{C}}=\{\ s^3\hat V_0+3s^2t\hat V_1+3st^2\hat V_2+t^3\hat V_3\ |\ s,t\in\mathbb{R}\}\mod X_F\] where \[\begin{aligned} &\hat V_0=\frac{3}{4}\partial_3,\\ &\hat V_1=\frac{1}{2}\partial_2+\frac{3}{8}\partial_3F\partial_3,\\ &\hat V_2= \frac{1}{2}\partial_1+\frac{1}{4}\partial_3F\partial_2+ \left(\frac{7}{20}\partial_2F -\frac{3}{20}X_F(\partial_3F) +\frac{9}{40}(\partial_3F)^2\right)\partial_3,\\ &\hat V_3=\partial_0+\frac{1}{4}\partial_3F\partial_1+ \left(\partial_2F-\frac{5}{4}X_F(\partial_3F)+\frac{7}{16}(\partial_3F)^2+\frac{7}{10}K\right)\partial_2\\ &\quad+\bigg(\partial_1F-\frac{3}{10}X_F(K)-X_F(\partial_2F)+\frac{21}{40}K\partial_3F\\ &\quad-\frac{27}{16}X_F(\partial_3F)\partial_3F-\frac{3}{4}\partial_2F\partial_3F+\frac{3}{4}X_F^2(\partial_3F)+\frac{27}{64}(\partial_3F)^3\bigg)\partial_3,\end{aligned}\] with \(K=-\partial_2F+\frac{3}{2}X(\partial_3F)-\frac{3}{8}(\partial_3F)^2\). To define the cone one looks for \((f,g)\) such that \[\tag{3.4} \mathrm{ad}_{fX_F}^4(g\partial_3)=0\mod X_F,\partial_3,\partial_2,\] where \(\mathrm{ad}^i_{X_F}\) stands for the iterated Lie bracket with the vector field \(X_F\). Then \(\hat{\mathcal{C}}_p\) is defined as the set of all \((\mathrm{ad}_{fX_F}^3(g\partial_3))(p)\), where \((f,g)\) solve (3.4). The explicit formula for \(\hat{\mathcal{C}}\) can be found using [12] and [12]. The cone \(\hat{\mathcal{C}}\) is invariant with respect to the flow of \(X_F\) if and only if (3.3) satisfies the Bryant–Wünschmann condition. In this case (3.4) takes the form \(\mathrm{ad}_{fX_F}^4(g\partial_3)=0\mod X_F\) (c.f. [13]). Then \(\hat{\mathcal{C}}\) can be projected to the quotient space \(J^3(\mathbb{R},\mathbb{R})/X_F\) and defines a \(\mathrm{GL}(2)\)-structure there via the field of cones \(\tilde{\mathcal{C}}=q_*\hat{\mathcal{C}}\), where \(q\colon J^3(\mathbb{R},\mathbb{R})\to J^3(\mathbb{R},\mathbb{R})/X_F\) is the quotient map. Note that \(J^3(\mathbb{R},\mathbb{R})/X_F\) can be identified with the hypersurface \(\{y=0\}\subset J^3(\mathbb{R},\mathbb{R})\). Denoting \[\begin{aligned} \alpha&=\partial_3F|_{y=0},\\ \beta&=\left(\frac{7}{20}\partial_2F-\frac{3}{20}X(\partial_3F)+\frac{9}{40}(\partial_3F)^2\right)\bigg|_{y=0},\\ \gamma&=\left(\partial_2F-\frac{5}{4}X_F(\partial_3F)+\frac{7}{16}(\partial_3F)^2+\frac{7}{10}K\right)\bigg|_{y=0},\\ \delta&=\left(\partial_1F-\frac{3}{10}X(K)-X(\partial_2F)+\frac{21}{40}K\partial_3F-\frac{27}{16}X(\partial_3F)\partial_3F\right.\\ &\phantom{=}\left.-\frac{3}{4}\partial_2F\partial_3F+\frac{3}{4}X^2(\partial_3F)+\frac{27}{64}(\partial_3F)^3\right)\bigg|_{y=0}\end{aligned}\] we get that \[\tilde{\mathcal{C}}=\{\ s^3V_0+3s^2tV_1+3st^2V_2+t^3V_3\ |\ s,t\in\mathbb{R}\}\] where \[\begin{aligned} &V_0=\frac{3}{4}\partial_3,\qquad V_1=\frac{1}{2}\partial_2+\frac{3}{8}\alpha\partial_3,\qquad V_2=\frac{1}{2}\partial_1+\frac{1}{4}\alpha\partial_2+\beta\partial_3,\\ &V_3=\partial_0+\frac{1}{4}\alpha\partial_1+\gamma\partial_2+\delta\partial_3. \end{aligned}\] The following linear change of coordinates \[(x_0,x_1,x_2,x_3)\mapsto\left(x_3, 2x_2, 2x_1, \frac{4}{3}x_0\right)\] transforms \((V_0,V_1,V_2,V_3)\) to \[\begin{aligned} &V_0=\partial_0,\qquad V_1=\partial_1+\frac{1}{2}\alpha\partial_0,\qquad V_2=\partial_2+\frac{1}{2}\alpha\partial_1+\frac{4}{3}\beta\partial_0,\\ &V_3=\partial_3+\frac{1}{2}\alpha\partial_2+2\gamma\partial_1+\frac{4}{3}\delta\partial_0, \end{aligned}\] which is equivalent to (3.2) up to constants.
Theorem 3.1 should be compared with [7], which can be rephrased that locally any torsion-free \(\mathrm{GL}(2)\)-structure admits a coframing of the form \(h\,\mathrm{d}x\) with \[\begin{aligned} &h=\\ &{\small\begin{pmatrix} a_1a_2a_3& a_0a_2a_3 & a_0a_1a_3 & a_0a_1a_2\\ \frac{1}{3}(a_1a_2b_3+a_1b_2a_3 & \frac{1}{3}(a_0a_2b_3+a_0b_2a_3 & \frac{1}{3}(a_0a_1b_3+a_0b_1a_2 & \frac{1}{3}(a_0a_1b_2+a_0b_1a_3 \\ +b_1a_2a_3) & +b_0a_2a_3) & +b_0a_1a_3) & +b_0a_1a_2) \\ \frac{1}{3}(a_1b_2b_3+b_1a_2b_3 & \frac{1}{3}(a_0b_2b_3+b_0a_2b_3 & \frac{1}{3}(a_0b_1b_3+b_0a_1b_3 & \frac{1}{3}(a_0b_1b_2+b_0a_1b_2 \\ +b_1b_2a_3) & +b_0b_2a_3) & +b_0b_1a_3) & +b_0a_1b_2) \\ b_1b_2b_3& b_0b_2b_3 & b_0b_1b_3 & b_0b_1b_2\end{pmatrix}}, \end{aligned}\] where \(a_i=\left(\frac{\partial u}{\partial x_i}\right)^{-1}\) and \(b_i=\left(\frac{\partial v}{\partial x_i}\right)^{-1}\) for some real-valued functions \(u\) and \(v\) on \(\mathcal{V}_3\simeq \mathbb{R}^4\). One checks that \(h\) is not contained in any proper subgroup of \(\mathrm{GL}(4,\mathbb{R})\). It is an interesting problem to find the smallest possible dimension of the group \(H\), such that all torsion-free \(\mathrm{GL}(2)\)-structures are \(H\)-flat (we believe that dimension 4 from Theorem 3.1 is optimal).