# zbMATH — the first resource for mathematics

Differential equations and conformal structures. (English) Zbl 1082.53024
This highly interesting work explores the deep geometrical relations between differential equations and conformal geometry. The first example of this fact was obtained by a student of Engel, K. Wünschmann, who defined conformal Lorentzian metrics on the solution space of a certain class of third order differential equations. This classical result was later reinterpreted in terms of Lie algebra valued connections.
This article presents various interesting examples pointing out the common aspects of differential equations and conformal structures. First, an alternative derivation of Wünschmann’s result is given, analyzing the properties for a third order equation to satisfy the so called Wünschmann condition. The conformal invariants of the corresponding Lorentzian metrics are computed. Further, new examples motivated by Cartan’s approach to the problem are given. More specifically, the author recovers Cartan’s observation that an additional condition on the Wünschmann class defines a 3-dimensional Lorentzian Weyl geometry satisfying the Einstein-Weyl equations. Two interesting examples on this fact are exhibited.
The most relevant part of the paper is devoted to the study of geometries related to underdetermined ordinary differential equations of Monge type and conformal geometries of signature $$(+++--)$$. Using the $$NG_{2}$$-valued Cartan connection, where $$NG_{2}$$ is the noncompact real form of the exceptional rank two Lie algebra $$G_{2}$$, the Cartan normal conformal connection for the geometry appearing in this type of equations is reduced. Another quite interesting point is the fact that the square of the Weyl tensor for the type of metrics analyzed can be interpreted by a classical invariant, which presents some affinities with polynomials used in the NP-formalism.

##### MSC:
 53B50 Applications of local differential geometry to the sciences 34A26 Geometric methods in ordinary differential equations 53B15 Other connections 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 58A15 Exterior differential systems (Cartan theory)
Full Text:
##### References:
 [1] Cartan, E., Sur LES espaces conformes generalises et l’universe optique, Comptes rendus acad. sci. Paris, 174, 857-859, (1921) · JFM 48.0854.04 [2] Cartan, E., Varietes a connexion projective, Bull. soc. math., LII, 205-241, (1924) · JFM 50.0500.02 [3] Cartan, E., La geometria de las ecuaciones diferenciales de tercer orden, Rev. mat. hispano-aamer., 4, 1-31, (1941) [4] Cartan, E., LES systemes de Pfaff a cinq variables et LES equations aux derivees partielles du second ordre, Ann. sc. norm. sup., 27, 109-192, (1910) · JFM 41.0417.01 [5] Chern, S.S., The geometry of the differential equations $$y''' = F(x, y, y^\prime, y'')$$, Sci. rep. nat. tsing hua univ., 4, 97-111, (1940) · JFM 66.0879.02 [6] Dunajski, M.; Mason, L.J.; Tod, K.P., Einstein – weyl geometry, the dkp equation and twistor theory, J. geom. phys., 37, 63-93, (2001) · Zbl 0990.53052 [7] Fefferman, C.L.; Fefferman, C.L., Monge – ampere equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. math., Ann. math., 104, 393-394, (1976) · Zbl 0332.32018 [8] Fritelli, S.; Kozameh, C.N.; Newman, E.T., GR via characteristic surfaces, J. math. phys., 36, 4984, (1995) · Zbl 0848.53045 [9] Fritelli, S.; Kozameh, C.; Newman, E.T., Differential geometry from differential equations, Commun. math. phys., 223, 383, (2001) · Zbl 1027.53080 [10] Fritelli, S.; Newman, E.T.; Nurowski, P., Conformal Lorentzian metrics on the spaces of curves and 2-surfaces, Class. Q. grav., 20, 3649-3659, (2003) · Zbl 1050.83021 [11] A.R Gover, P. Nurowski, Obstructions to conformally Einstein metrics, 2004. math.DG/0405304 at http://xxx.lanl.gov. · Zbl 1098.53014 [12] Hilbert, D., Ueber den begriff der klasse von differentialgleichungen, Mathem. annalen bd., 73, 95-108, (1912) · JFM 43.0378.01 [13] M. Godlinski, P. Nurowski, Differential geometry of the third order ODEs, in preparation. · Zbl 1098.34005 [14] Kobayashi, S., Transformation groups in differential geometry, (1972), Springer Berlin · Zbl 0246.53031 [15] Newman, E.T.; Nurowski, P., Projective connections associated with second order odes, Class. Q. grav., 20, 2325-2335, (2003) · Zbl 1045.53013 [16] Newman, E.T.; Penrose, R., An approach to gravitational radiation by a method of spin coefficients, J. math. phys., 3, 566, (1962) · Zbl 0108.40905 [17] Nurowski, P.; Sparling, G.A.J., Three-dimensional cauchy – riemann structures and second order ordinary differential equations, Class. Q. grav., 20, 4995-5016, (2003) · Zbl 1051.32019 [18] Olver, P.J., Equivalence invariants and symetry, (1996), Cambridge University Press Cambridge [19] Penrose, R., A spinor approach to general relativity, Ann. phys. (NY), 10, 171-201, (1960) · Zbl 0091.21404 [20] Penrose, R., Twistor algebra, J. math. phys., 8, 345-366, (1967) · Zbl 0163.22602 [21] Petrov, A.Z., Classification of spaces defining gravitational fields, Sci. not. kazan state univ., 114, 55-69, (1954) [22] G.A.J. Sparling, Private communications, 2003. [23] Tod, K.P., Einstein – weyl spaces and third order differential equations, J. math. phys., 41, 5572, (2000) · Zbl 0979.53050 [24] Wuenschmann, K., Ueber beruhrungsbedingungen bei differentialgleichungen, (1905), Dissertation Greifswald
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.