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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