New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions

Joël Merker ; Paweł Nurowski

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

On a $3$D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, $1$-form) modulo gauge $\widehat{g} = {\rm e}^{2\varphi} g$, $\widehat{A} = A + {\rm d}\varphi$. In 1943, Cartan showed that every solution to the Einstein–Weyl equations $R_{(\mu\nu)} - \frac{1}{3} R g_{\mu\nu} = 0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3$^{\mathrm{rd}}$ order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided $2$ among $3$ primary point invariants vanish \begin{gather*} \text{Wunschmann}(H) \equiv 0\equiv \text{Cartan}(H). \end{gather*} We find that point equivalence of a single PDE $z_y = F(x,y,z,z_x)$ with para-CR integrability $DF := F_x + z_x F_z \equiv 0$ leads to a completely similar $7$D Cartan bundle and connection. Then magically, the (complicated) equation $\text{Wunschmann}(H) \equiv 0$ becomes \begin{gather*} 0\equiv\text{Monge}(F):=9F_{pp}^2F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^3,\qquad p:=z_x, \end{gather*} whose solutions are just conics in the $\{p, F\}$-plane. As an ansatz, we take \begin{gather*} F(x,y,z,p):= \frac{\alpha(y)(z-xp)^2\!+\beta(y)(z-xp)p+\gamma(y)(z-xp) +\delta(y)p^2\!+\varepsilon(y)p+\zeta(y)}{\lambda(y)(z-xp)+\mu(y) p+\nu(y)},\! \end{gather*} with $9$ arbitrary functions $\alpha, \dots, \nu$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H) \equiv 0 $ passes to a certain $\text{Cartan}(F) \equiv 0$ which holds for any choice of $\alpha(y), \dots, \nu(y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein–Weyl structures $\big[ (g, A) \big]$ in $3$D. These structures are nontrivial in the sense that $\mathrm{d}A \not\equiv 0$ and $\text{Cotton}([g]) \not \equiv 0$.

Détail

Keywords: Einstein–Weyl structures, Lorentzian metrics, third-order ordinary differential equations, Cartan's method of equivalence, exterior differential systems.
Mots-clés : para-CR structures, Monge invariant, Wünschmann invariant

Bibliographie

[1] Calderbank D. M.J., Pedersen H., “Einstein–Weyl geometry”, Surveys in Differential Geometry: Essays on Einstein Manifolds, Surv. Differ. Geom., 6, Int. Press, Boston, MA, 1999, 387–423 | DOI | MR | Zbl

[2] Cartan E., “La geometria de las ecuaciones diferenciales de tercer orden”, Rev. Mat. Hispano-Amer., 4 (1941), 1–31

[3] Cartan E., “Sur une classe d'espaces de Weyl”, Ann. Sci. École Norm. Sup., 60 (1943), 1–16 | DOI | MR | Zbl

[4] Dunajski M., Mason L. J., Tod P., “Einstein–Weyl geometry, the dKP equation and twistor theory”, J. Geom. Phys., 37 (2001), 63–93, arXiv: math.DG/0004031 | DOI | MR | Zbl

[5] Eastwood M. G., Tod K. P., “Local constraints on Einstein–Weyl geometries: the 3-dimensional case”, Ann. Global Anal. Geom., 18 (2000), 1–27 | DOI | MR | Zbl

[6] Frittelli S., Kozameh C., Newman E. T., “Differential geometry from differential equations”, Comm. Math. Phys., 223 (2001), 383–408, arXiv: gr-qc/0012058 | DOI | MR | Zbl

[7] Godlinski M., Geometry of third-order ordinary differential equations and its applications in general relativity, arXiv: 0810.2234

[8] Godlinski M., Nurowski P., Geometry of third order ODEs, arXiv: 0902.4129

[9] Hill C. D., Nurowski P., “Differential equations and para-CR structures”, Boll. Unione Mat. Ital., 3 (2010), 25–91, arXiv: 0909.2458 | MR | Zbl

[10] Hitchin N. J., “Complex manifolds and Einstein's equations”, Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., 970, Springer, Berlin – New York, 1982, 73–99 | DOI | MR

[11] Jones P. E., Tod K. P., “Minitwistor spaces and Einstein–Weyl spaces”, Classical Quantum Gravity, 2 (1985), 565–577 | DOI | MR | Zbl

[12] LeBrun C., Mason L. J., “The Einstein–Weyl equations, scattering maps, and holomorphic disks”, Math. Res. Lett., 16 (2009), 291–301, arXiv: 0806.3761 | DOI | MR | Zbl

[13] Merker J., “Lie symmetries and CR geometry”, J. Math. Sci., 154 (2008), 817–922, arXiv: math.CV/0703130 | DOI | MR | Zbl

[14] Merker J., Nurowski P., On degenerate para-CR structures: Cartan reduction and homogeneous models, arXiv: 2003.08166

[15] Merker J., Pocchiola S., “Explicit absolute parallelism for 2-nondegenerate real hypersurfaces $M^5 \subset {\mathbb C}^3$ of constant Levi rank 1”, J. Geom. Anal., 30 (2020), 2689–2730 | DOI | MR | Zbl

[16] Monge G., “Sur les équations différentielles des courbes du second degré”, Corr. l'École Impériale Polytech., 1810, no. 2, 51–54

[17] Nurowski P., “Differential equations and conformal structures”, J. Geom. Phys., 55 (2005), 19–49, arXiv: math.DG/0406400 | DOI | MR | Zbl

[18] Pedersen H., Tod K. P., “Three-dimensional Einstein–Weyl geometry”, Adv. Math., 97 (1993), 74–109 | DOI | MR | Zbl

[19] Tod K. P., “Compact $3$-dimensional Einstein–Weyl structures”, J. London Math. Soc., 45 (1992), 341–351 | DOI | MR | Zbl

[20] Tod K. P., “Einstein–Weyl spaces and third-order differential equations”, J. Math. Phys., 41 (2000), 5572–5581 | DOI | MR | Zbl

[21] Weyl H., Raum, Zeit, Materie, Springer-Verlag, Berlin, 1919 | DOI | MR | Zbl

[22] Wünschmann K., Uber Berührungsbedingungen bei Integralkurven von Differentialgleichungen, Inaug. Dissert., Teubner, Leipzig, 1905 | Zbl