Riv. Mat. Univ. Parma, Vol. 11, No. 1, 2020

Elisabetta Barletta [a] and Sorin Dragomir [a]

Robinson-Sparling construction of CR structures associated to shearfree null geodesic congruences

Pages: 9-68
Received: 14 January 2019
Accepted in revised form: 7 May 2019
Mathematics Subject Classification (2010): 32V20, 32V30, 53C50, 53D10, 53Z05.
Keywords: CR structure, tangential Cauchy-Riemann equations, Fefferman metric, flag structure, optical structure, Lorentzian metric, space-time, Maxwell field, Dirac equation.
Authors address:
[a]: Università degli Studi della Basilicata, Dipartimento di Matematica, Informatica ed Economia, Via dell'Ateneo Lucano 10, Potenza, 85100, Italy

Full Text (PDF)

Abstract: We review the construction of Lorentzian metrics, such as Fefferman type metrics, associated to a given \(3\)-dimensional nondegenerate CR manifold \(M\), and admitting shearfree null geodesic congruences \(N\). This class of metrics is obtained by a lifting procedure from \(M\) to \(M \times {\mathbb R}\) devised by I. Robinson and A. Trautman (cf. [71]-[72]) and notably radiative gravitational fields are searched for (cf. e.g. R.K. Sachs, [74]) within the class. Conversely, nondegenerate CR structures arise (by the Robinson-Trautmann construction, [71]) on leaf spaces \({\mathfrak M}/N\) associated to space-times \(\mathfrak M\) adapted to given optical structures \(\big( (K, L), \, J)\). The Graham-Sparling construction (cf. [40], [77]) is shown to be a particular case of Robinson-Trautman construction where the complex structure on the complex line bundle \({\rm Ker} (L) /K \to {\mathfrak M}\) is induced by an \(f\)-structure with two complemented frames obtained as a covariant derivative of the given null Killing vector field \(N\).

References
[1]
T. Akahori, A new approach to the local embedding theorem of CR-structures for \(n\geq4\) (the local solvability for the operator \(\overline\partial_b\) in the abstract sense), Mem. Amer. Math. Soc. 67 (1987), no. 366. MR0888499
[2]
A. Andreotti and C. D. Hill, Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Scuola Norm. Sup. Pisa 26 (1972), 299-324. MR0460724
[3]
A. Andreotti and G. A. Fredricks, Embeddability of real analytic Cauchy-Riemann manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6 (1979), 285-304. MR0541450
[4]
F. Antonacci and P. Piccione, A Fermat principle on Lorentzian manifolds and applications, Appl. Math. Lett. 9 (1996), 91-95. MR1383689
[5]
M. Arminjon and F. Reifler, Four-vector versus four-scalar representation of the Dirac wave function, Int. J. Geom. Methods Mod. Phys. 9 (2012), 1250026, 23 pp. MR2917305
[6]
M. Arminjon and F. Reifler, Basic quantum mechanics for three Dirac equations in a curved spacetime, Braz. J. Phys. 40 (2010), 242-255. DOI
[7]
M. Arminjon and F. Reifler, Equivalent forms of Dirac equations in curved space-times and generalized de Broglie relations, Braz. J. Phys. 43 (2013), 64-77. DOI
[8]
A. Banerjee, Null electromagnetic fields in general relativity, J. Phys. A: Gen. Phys. 3 (1970), 501-504. DOI
[9]
E. Barletta and S. Dragomir, On the CR structure of the tangent sphere bundle, Matematiche (Catania) 50 (1995), 237-249. MR1414632
[10]
E. Barletta and S. Dragomir, Transversally CR foliations, Rend. Mat. Appl. (7) 17 (1997), 51-85. MR1459408
[11]
E. Barletta and S. Dragomir, New CR invariants and their application to the CR equivalence problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1997), 193-203. MR1475776
[12]
E. Barletta and S. Dragomir, Differential equations on contact Riemannian manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30 (2001), 63-95. MR1882025
[13]
E. Barletta, S. Dragomir and K. L. Duggal, Foliations in Cauchy-Riemann geometry, Math. Surveys Monogr., 140, American Mathematical Society, Providence, 2007. MR2319199
[14]
E. Barletta, S. Dragomir and H. Jacobowitz, Gravitational field equations on Fefferman space-times, Complex Anal. Oper. Theory 11 (2017), 1685-1713. MR3717385
[15]
H. Bateman, The transformations of coordinates which can be used to transform one physical problem into another, Proc. London Math. Soc. 8 (1910), 469-488. MR1577445
[16]
S. Bergman, The kernel function and conformal mapping, Mathematical Surveys, No. 5, American Mathematical Society, New York, 1950. MR0038439
[17]
D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR0467588
[18]
D. E. Blair, Geometry of manifolds with structural group \({\rm U}(n) \times {\rm O}(s)\), J. Differential Geometry 4 (1970), 155-167. MR0267501
[19]
D. E. Blair, On a generalization of the Hopf fibration, An. Şti. Univ. ''Al. I. Cuza'' Iaşi Secţ. I a Mat. (N.S.) 17 (1971), 171-177. MR0314064
[20]
D. E. Blair, G. D. Ludden and K. Yano, Differential geometric structures on principal toroidal bundles, Trans. Amer. Math. Soc. 181 (1973), 175-184. MR0319099
[21]
H. Bondi, F. A. E. Pirani and I. Robinson, Gravitational waves in general relativity, III, Exact plane waves, Proc. Roy. Soc. London Ser. A 251 (1959), 519-533. MR0106747
[22]
H. Bondi and F. A. E. Pirani, Gravitational waves in general relativity, XIII, Caustic property of plane waves, Proc. Roy. Soc. London Ser. A 421 (1989), 395-410. MR0985269
[23]
W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. 68 (1958), 721-734. MR0112160
[24]
S.-S. Chern, On a generalization of Kähler geometry, Algebraic Geometry and Topology, Sympos. in honor of S. Lefschetz, Princeton Univ. Press, Princeton, 1957, 103-121. MR0087172
[25]
S. Dain and O. M. Moreschi, The Goldberg-Sachs theorem in linearized gravity, J. Math. Phys. 41 (2000), 6296-6299. MR1779646
[26]
R. Debever, Sur les espaces de Brandon Carter, Acad. Roy. Belg. Bull. Cl. Sci. 55 (1969), 8-16. MR0247860
[27]
P. A. M. Dirac, The principles of quantum mechanics, \(4^{\rm th}\) ed., Oxford University Press, Oxford, 1959. MR0023198
[28]
P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73-88. MR0141050
[29]
S. Dragomir, Cauchy-Riemann submanifolds of Kaehlerian Finsler spaces, Collect. Math. 40 (1989), 225-240. MR1099243
[30]
S. Dragomir and J. Masamune, Cauchy-Riemann orbifolds, Tsukuba J. Math. 26 (2002), 351-386. MR1940400
[31]
S. Dragomir and G. Tomassini, Differential geometry and analysis on CR manifolds, Progress in Mathematics, 246, Birkhäuser, Boston, 2006. MR2214654
[32]
C. Fefferman, The Bergman kernel and biholomorphic equivalence of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. MR0350069
[33]
C. Fefferman, Monge-Ampére equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395-416; ibidem, 104 (1976), 393-394. MR0407320
[34]
R. Geroch, Partial differential equations of physics, in ''General relativity'' (Aberdeen, 1995), Scott. Univ. Summer School Phys., Edinburgh, 1996, 19-60. MR1412628
[35]
F. Giannoni and A. Masiello, On a Fermat principle in general relativity. A Morse theory for light rays, Gen. Relativity Gravitation 28 (1996), 855-897. MR1398288
[36]
K. Gödel, An example of a new type of cosmological solutions of Einstein's field equations of gravitation, Rev. Modern Physics 21 (1949), 447-450. MR0031841
[37]
S. I. Goldberg, A generalization of Kaehler geometry, J. Differential Geometry 6 (1972), 343-355. MR0300227
[38]
J. N. Goldberg and R. K. Sachs, A theorem on Petrov types, Acta Phys. Polon. 22 (1962), 13-23. MR0156679
[39]
S. I. Goldberg and K. Yano, On normal globally framed \(f\)-manifolds, Tohoku Math. J. 22 (1970), 362-370. MR0305295
[40]
C. R. Graham, On Sparling's characterization of Fefferman metrics, American J. Math. 109 (1987), 853-874. MR0910354
[41]
S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press, New York-London, 1978. MR0514561
[42]
C. D. Hill, J. Lewandowski and P. Nurowski, Einstein's equations and the embedding of \(3\)-dimensional CR manifolds, Indiana Univ. Math. J. 57 (2008), 3131-3176. MR2492229
[43]
J. Holland and G. Sparling, Null electromagnetic fields and relative Cauchy-Riemann embeddings, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2016), 20120583, 14 pp. MR3016190 | DOI
[44]
S. Ianuş, Sulle varietà di Cauchy-Riemann, Rend. Accad. Sci. Fis. Mat. Napoli 39 (1972), 191-195. MR0343204
[45]
F. John, Partial differential equations, 4th ed., Appl. Math. Sci., 1, Springer-Verlag, New York, 1982. MR0831655
[46]
N. Kamran and R. G. McLenaghan, Separation of variables and symmetry operators for the neutrino and Dirac equations in the space-times admitting a two-parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences, J. Math. Phys. 25 (1984), 1019-1027. MR0739257
[47]
S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Publishers, New York, Vol. I, 1963; Vol. II, 1969. MR0152974 |  MR0238225
[48]
L. K. Koch, Chains on CR manifolds and Lorentz geometry, Trans. Amer. Math. Soc. 307 (1988), 827-841. MR0940230
[49]
L. K. Koch, Chains, null-chains, and CR geometry, Trans. Amer. Math. Soc. 338 (1993), 245-261. MR1100695
[50]
W. Kofink, Zur Mathematik der Diracmatrizen: Die Bargmannsche Hermitisierungsmatrix A und die Paulische Transpositionsmatrix B, Math. Z. 51 (1949), 702-711. MR0032563
[51]
A. Korányi and H. M. Reimann, Contact transformations as limits of symplectomorphisms, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 1119-1124. MR1282355
[52]
M. Kuranishi, Strongly pseudoconvex CR structures over small balls, I, II, III, Ann. of Math. 115 (1982), 451-500; ibidem 116 (1982), 1-64; ibidem 116 (1982), 249-330. MR0657236
[53]
J. M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), 411-429. MR0837820
[54]
J. M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), 157-178. MR0926742
[55]
T. Levi-Civita, Sulle funzioni di due o più variabili complesse, Atti Accad. Naz. Lincei Rend. (V) 14 (1905), 492-499. zbMATH |  Article
[56]
A. Lichnerowicz, Propagateurs, commutateurs et anticommutateurs en relativité générale, in ''Relativité, Groupes et Topologie'', (Les Houches Lectures, 1963), B. DeWitt and C. DeWitt, eds., Gordon and Breach, New York, 1964, 821-861. MR0168336 |  MR0168328
[57]
A. Masiello and P. Piccione, Shortening null geodesics in Lorentzian manifolds. Applications to closed light rays, Differential Geom. Appl. 8 (1998), 47-70. MR1601534
[58]
E. Musso, The local embedding problem for optical structures, J. Geom. Phys. 10 (1992), 1-18. MR1195669
[59]
P. Molino, Riemannian foliations, Progr. Math., 73, Birkhäuser, Boston, 1988. MR0932463
[60]
A. Morimoto, On normal almost contact structures with a regularity, Tohoku Math. J. 16 (1964), 90-104. MR0163246
[61]
L. Nirenberg, On a question of Hans Lewy, Russian Math. Surveys 29 (1974), 251-262. MR0492752
[62]
M. Novello and J. Duarte De Oliveira, On dual properties of the Weyl tensor, Gen. Relativity Gravitation 12 (1980), 871-880. MR0601845
[63]
M. Ortaggio, Bel-Debever criteria for the classification of the Weyl tensor in higher dimensions, Classical Quantum Gravity 26 (2009), 195015, 8 pp. MR2545152 | DOI
[64]
R. S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. No. 22, 1957. MR0121424
[65]
W. Pauli, Contributions mathématiques à la théorie des matrices de Dirac, (French), Ann. Inst. H. Poincaré 6 (1936), 109-136. MR1508031
[66]
A. Z. Petrov, Klassifikacya prostranstv opredelyayushchikh polya tyagoteniya, (Russian), Uch. Zapiski Kazan. Gos. Univ. 114 (1954), 55-69. English translation: The classification of spaces defining gravitational fields, Gen. Relativity Gravitation 32 (2000), 1665-1685. MR1784371 | DOI
[67]
F. Pirani, On the physical significance of the Riemann tensor, Acta Phys. Polon. 15 (1956), 389-405. MR0088370
[68]
F. Pirani, Invariant formulation of gravitational radiation theory, Phys. Rev. 105 (1957), 1089-1099. MR0096537
[69]
I. Robinson, Null electromagnetic fields, J. Mathematical Phys. 2 (1961), 290-291. MR0127369
[70]
I. Robinson and A. Trautman, Conformal geometry of flows in \(n\) dimensions, J. Math. Phys. 24 (1983), 1425-1429. MR0708658
[71]
I. Robinson and A. Trautman, Cauchy-Riemann structures in optical geometry, in ''Proc. fourth Marcel Grossmann meeting on general relativity'', R. Ruffini, ed., North-Holland Publishing, Amsterdam, 1986, 317-324. MR0879758
[72]
I. Robinson and A. Trautman, Optical geometry, in ''New Theories in Physics'', Z. Ajduk, S. Pokorski and A. Trautman, eds., World Scientific, 1989, 454-497. WorldCat
[73]
R. Sachs, Gravitational waves in general relativity. VI. The outgoing radiation condition, Proc. Roy. Soc. Ser. A 264 (1961), 309-338. MR0156678
[74]
R. K. Sachs, Gravitational radiation, in ''Recent developments in general relativity'', Pergamon, 1962, 521-562. MR0164694
[75]
M. Sánchez, Lorentzian manifolds admitting a Killing vector field, Nonlinear Anal. 30 (1997), 643-654. MR1489831
[76]
I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359-363. MR0079769
[77]
G. Sparling, Twistor theory and the characterization of Fefferman's conformal structures, preprint.
[78]
N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Lecture in Math., Dept. of Mathematics, Kyoto Univ., No. 9, Kinokuniya Book Store Co., Tokyo, 1975. MR0399517 | PDF
[79]
S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349-379. MR1000553
[80]
Ph. Tondeur, Foliations on Riemannian manifolds, Universitext, Springer-Verlag, New York, 1988. MR0934020
[81]
A. Trautman, Optical structures in relativistic theories, The mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque 1985, 401-420. MR0837209
[82]
A. Trautman, Deformation of the Hodge map and optical geometry, J. Geom. Phys. 1 (1984), 85-95. MR0794981
[83]
A. Trautman, Robinson manifolds and Cauchy-Riemann spaces, Classical Quantum Gravity 19 (2002), R1-R10. MR1885472
[84]
S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25-41. MR0520599
[85]
K. Yano, On a structure defined by a tensor field \(f\) of type \((1,1)\) satisfying \(f^3 + f = 0\), Tensor (N.S.) 14 (1963), 99-109. MR0159296
[86]
K. Yano and S. Ishihara, On integrability conditions of a structure \(f\) satisfying \(f^3 + f = 0\), Quart. J. Math. Oxford Ser. (2) 15 (1964), 217-222. MR0166718
[87]
X. Zhang and D. Finley, CR structures and twisting vacuum spacetimes with two Killing vectors and cosmological constant: type II and more special, Classical Quantum Gravity 30 (2013), 115006, 20 pp. MR3055095


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