The Cayley Cubic and Differential Equations

Abstract

We define Cayley structures as a field of Cayley’s ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations and the theory of integrable systems. In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair. After solving the equivalence problem we obtain the fundamental invariants, find the local generality of several classes of Cayley structures and give examples.

Appendix

In this section we will present the relations between the quantities $b_{ijk}$ in structure equations (2.15) when $a_7 = 2a_6$ in the branching (2.16) as described below

$$\begin{aligned} \begin{aligned}&b_{104} = -a_4,\ \ b_{105} = -a_2, \ \ b_{113} = b_{223},\ \ b_{114} = -2a_3,\ \ b_{115} = a_1, \ \ b_{124} = -a_1,\ \ b_{125} = 0,\\&b_{134} = 0,\ \ b_{135} = 0,\ \ b_{145} = 0,\ \ b_{203} = b_{102},\ \ b_{204} = -a_6, b_{205} = 0,\ \ b_{215} = -a_3+a_2, \\&b_{224} = -2a_3,\ \ b_{225} = a_1,\ \ b_{235}=0,\ \ b_{245} = 0,\ \ b_{302}= -b_{402}+b_{201},\ \ b_{303} = -b_{403}+b_{101},\\&b_{304} = 0,\ \ b_{305} = -5a_6,\ \ b_{312} = b_{202}-b_{101},\ \ b_{313}=b_{102},\ \ b_{314} = a_6,\ \ b_{315} = 0,\\&b_{323} = b_{103},\ \ b_{324} = -a_4,\ \ b_{325} = -a_2,\ \ b_{334} = a_2,\ \ b_{335} = 0,\ \ b_{345}=0,\ \ b_{404} = 0,\\&b_{405} = 3a_6,\ \ b_{413} = \textstyle {\frac{1}{2}}(b_{212}-b_{102}),\ \ b_{414} = \textstyle {-\frac{3}{2}}a_6,\ \ b_{415}=-\textstyle {\frac{1}{2}}a_5,\\&b_{423} = -b_{103}-b_{112},\ \ b_{424} = a_5+a_4,\ \ b_{425} = -a_3+2a_2,\\&b_{434} = -a_3-a_2, b_{435} = a_1,\ \ b_{445} = 0. \end{aligned} \end{aligned}$$

(A.1)

For the branch $a_7 = \textstyle {\frac{3}{2}}a_6$, in (2.16), the quantities that are different from the ones in (A.1) are

$$\begin{aligned} b_{305}=-\textstyle {\frac{7}{2}a_6}, \ \ b_{314}=\textstyle {\frac{1}{2}a_6},\ \ b_{405}=-\textstyle {2a_6},\ \ b_{414}=-a_6 \end{aligned}$$

Additionally, the expression of the $\mathfrak {sl}_4(\mathbb {R})$-valued Cartan connection (3.2), mentioned in Sect. 4.3, satisfying (4.11), is as follows when $a_7 = 2a_6$ in the branching (2.16).

$$\begin{aligned} \begin{aligned} \psi _0&= -(a_4 + a_5) \omega ^0 + \phi _0,\\ \psi _1&= -(a_4 + a_5) \omega ^0+(a_2 - a_3) \omega ^1 + \phi _0 + \phi _1,\\ \psi _2&= - a_5 \omega ^0+a_1 \omega ^2 - 2 \phi _1,\\ \gamma _1&=\textstyle {\frac{1}{2} a_6 \omega ^0-(a_4+a_5)\omega ^1+(-\frac{3}{4}a_2+\frac{1}{2}a_3) \omega ^2},\\ \gamma _2&= \textstyle {2 a_6 \omega ^1-a_4 \omega ^2+a_2 \omega ^3,}\\ \mu _3&=\textstyle {(- \frac{3}{4} b_{101}-\frac{3}{4}b_{403}+\frac{1}{4} b_{412}+\frac{1}{4} a_{3;2}-\frac{1}{4} a_{4;3}+\frac{1}{2} a_1 a_2) \omega ^0-b_{212} \omega ^1}\\&+ \textstyle {(\frac{5}{4}b_{112}-\frac{1}{4} a_{1;3}+\frac{3}{4} b_{103}) \omega ^2+b_{223} \omega ^3-2 a_3 \theta ^1+a_1 \theta ^2},\\ \mu _2&=\textstyle {(\frac{3}{2} a_3 a_4+\frac{3}{2} a_5 a_3-\frac{1}{4} a_1 a_6-\frac{9}{4} a_4 a_2-\frac{45}{16} a_2 a_5-\frac{5}{16}b_{301}-\frac{7}{8} b_{401}-\frac{7}{8} a_{4;1}-a_{5;1}) \omega ^0}\\&+\textstyle {b_{201} \omega ^1+\frac{1}{4}(2 a_1 a_2+a_{3;2}-3 b_{101}+4b_{202}-3 b_{403}+b_{412}- a_{4;3}) \omega ^2+b_{102} \omega ^3-a_6 \theta ^1},\\ \mu _1&=\textstyle {(-\frac{1}{2} a_2a_3 +\frac{3}{4} a_2^2-\frac{1}{2} a_{6;3})\omega ^0+\frac{1}{4}(2 a_1a_2- b_{101}+ 3a_{3;2} + a_{4;3}- b_{403}+ 3 b_{412})\omega ^1}\\&+\textstyle {\frac{1}{4}(3b_{102}+ b_{212}- a_{2;3})\omega ^2+\frac{1}{4}(b_{103}-b_{112}+a_{1;3})\omega ^3+(-a_4-a_5)\theta ^1+(-\frac{3}{4} a_2+\frac{1}{2} a_3)\theta ^2}\\ \mu _0&=\textstyle {(4a_2a_6-\frac{5}{2} a_3a_6+4a_4a_5+2a_4^2+2a_5^2-\frac{1}{2} a_{6;1})\omega ^0}\\&+\textstyle {(-\frac{3}{4}a_1a_6-\frac{3}{4} a_4a_2-\frac{15}{16}a_2a_5+\frac{1}{2}a_3a_4+\frac{1}{2} a_5a_3+\frac{9}{16}b_{301}+\frac{3}{8}b_{401}+\frac{3}{8}a_{4;1})\omega ^1}\\&+\textstyle {\frac{1}{4}(-2a_2a_3+ 3a_2^2+b_{201}- 3b_{402}- a_{6;3}+ a_{4;2}+ a_{5;2})\omega ^2}\\&+\textstyle {\frac{1}{4}(-2 a_1a_2+ 3 b_{101}- b_{403}- b_{412}- a_{3;2}+ a_{4;3})\omega ^3-\frac{5}{2} a_6\theta ^2}\\ \end{aligned} \end{aligned}$$

(A.2)

For the branch $a_7 = \textstyle {\frac{3}{2}}a_6$ let us denote the entries of the $\mathfrak {sl}_4(\mathbb {R})$-valued Cartan connection (3.2) not included in (4.11) by $\tilde{\psi }_0,\tilde{\psi }_1,\tilde{\psi }_2,\tilde{\gamma }_1,\tilde{\gamma }_2,\tilde{\mu }_3,\tilde{\mu }_2,\tilde{\mu }_1,\tilde{\mu }_0.$ Using the expressions given in (A.2) one obtains

$$\begin{aligned} \begin{aligned}&\tilde{\psi }_0=\psi _0,\qquad \tilde{\psi }_1=\psi _1,\qquad \tilde{\psi }_2=\psi _2,\qquad \tilde{\gamma }_1=\gamma _1-\textstyle {\frac{1}{8}}a_6\omega ^0,\qquad \tilde{\gamma }_2=\gamma _2-\textstyle {\frac{1}{2}}a_6\omega ^1,\\&\tilde{\mu }_3=\mu _3,\qquad \tilde{\mu }_2=\mu _2+\textstyle {\frac{1}{16}}(a_1a_6-3a_2a_5-3b_{301}-6b_{401})\omega ^0,\qquad \tilde{\mu }_1=\mu _1-\textstyle {\frac{1}{8}}a_{6;3}\omega ^0,\\&\tilde{\mu }_0=\mu _0 +\frac{1}{8}(-20a_6 a_2+15a_3 a_6-5 a_{6;1}) \omega ^0\\&+\frac{1}{16}(3a_1 a_6- a_2 a_5- b_{301}-2b_{401}) \omega ^1-\frac{1}{8}a_{6;3} \omega ^2 +\frac{11}{8}a_6 \theta ^2 \end{aligned} \end{aligned}$$

(A.3)