zbMATH — the first resource for mathematics

Differential invariants of Einstein-Weyl structures in 3D. (English) Zbl 1392.53054
Summary: Einstein-Weyl structures on a three-dimensional manifold \(M\) are given by a system \(\mathcal{E}\) of PDEs on sections of a bundle over \(M\). This system is invariant under the Lie pseudogroup \(\mathcal{G}\) of local diffeomorphisms on \(M\). Two Einstein-Weyl structures are locally equivalent if there exists a local diffeomorphism taking one to the other. Our goal is to describe the quotient equation \(\mathcal{E} / \mathcal{G}\) whose solutions correspond to nonequivalent Einstein-Weyl structures. The approach uses symmetries of the Manakov-Santini integrable system and the action of the corresponding Lie pseudogroup.
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
Full Text: DOI
[1] Weyl, H., Reine infinitesimalgeometrie, Math. Z., 2, 384-411, (1918) · JFM 46.1301.01
[2] Calderbank, D. M.J., Integrable background geometries, SIGMA, 10, 034, (2014) · Zbl 1288.53074
[3] Ferapontov, E. V.; Kruglikov, B. S., Dispersionless integrable systems in 3D and Einstein-Weyl geometry, J. Differential Geom., 97, 215-254, (2014) · Zbl 1306.37084
[4] Hitchin, N. J., Complex manifolds and einstein’s equations, (Twistor Geometry and Non-Linear Systems, Lecture Notes in Math., vol. 970, (1982), Springer) · Zbl 0507.53025
[5] Jones, P. E.; Tod, K. P., Minitwistor spaces and Einstein-Weyl spaces, Classical Quantum Gravity, 2, 4, 565-577, (1985) · Zbl 0575.53042
[6] Dunajski, M.; Mason, L. J.; Tod, P., Einstein-Weyl geometry the dkp equation and twistor theory, J. Geom. Phys., 37, 1-2, 63-93, (2001) · Zbl 0990.53052
[7] Cartan, E., Sur une classe d’espaces de Weyl, Ann. Sci. École Norm. Sup. (3), 60, 1-16, (1943) · Zbl 0028.30802
[8] Thomas, T. Y., The differential invariants of generalized spaces, (1934), Cambridge University Press Cambridge · JFM 60.0363.02
[9] Alekseevskij, D.; Lychagin, V.; Vinogradov, A., Basic ideas and concepts of differential geometry, Geometry, vol. 1, Encycl. Math. Sci., 28, (1991)
[10] Olver, P., Equivalence, invariants, and symmetry, (1995), Cambridge University Press Cambridge · Zbl 0837.58001
[11] Dunajski, M.; Ferapontov, E. V.; Kruglikov, B., On the Einstein-Weyl and conformal self-duality equations, J. Math. Phys., 56, (2015) · Zbl 1325.53058
[12] Manakov, S. V.; Santini, P. M., Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation, JETP Lett., 83, 462-466, (2006)
[13] Krasilshchik, I.; Lychagin, V.; Vinogradov, A., Geometry of jet spaces and nonlinear partial differential equations, (1986), Gordon and Breach
[14] Kruglikov, B.; Lychagin, V., Geometry of differential equation, (Krupka, D.; Saunders, D., Handbook of Global Analysis, Vol. 72, (2008), Elsevier), 5-772
[15] Kruglikov, B.; Lychagin, V., Global Lie-tresse theorem, Selecta Math., 22, 1357-1411, (2016) · Zbl 1347.53015
[16] Kruglikov, B.; Schneider, E., Differential invariants of self-dual conformal structures, J. Geom. Phys., 113, 176-187, (2017) · Zbl 1360.53024
[17] Lychagin, V.; Yumaguzhin, V., Invariants in relativity theory, J. Math., 36, 3, 298-312, (2015) · Zbl 1332.83034
[18] Calabi, E.; Olver, P.; Shakiban, C.; Tannenbaum, A.; Haker, S., Differential and numerically invariant signature curves applied to object recognition, Int. J. Comput. Vis., 26, 2, 107-135, (1998)
[19] Khokhlov, R. V.; Zabolotskaya, E. A., Quasi-plane waves in the nonlinear acoustics of confined beam, Sov. Phys. Acoust., 15, 35-40, (1969)
[20] Kadomtsev, B. B.; Petviashvili, V. I., On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15, 539-541, (1970) · Zbl 0217.25004
[21] Martínez Alonso, L.; Shabat, A. B., Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type, Phys. Lett. A, 300, 54-58, (2002) · Zbl 0997.37045
[22] E.V. Ferapontov, B.S. Kruglikov, Dispersionless integrable hierarchies and GL(2,R) geometry, 2016, arXiv:1607.01966.
[23] Ferapontov, E. V.; Khusnutdinova, K. R., On the integrability of (2+1)-dimensional quasilinear systems, Comm. Math. Phys., 248, 187-206, (2004) · Zbl 1070.37047
[24] Anderson, I. M.; Kruglikov, B., Rank 2 distributions of Monge equations: symmetries, equivalences, extension, Adv. Math., 228, 3, 1435-1465, (2011) · Zbl 1234.34010
[25] Anderson, I. M.; Kamran, N.; Olver, P., Internal, external and generalized symmetrie, Adv. Math., 100, 53-100, (1993) · Zbl 0809.58044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.