Abstract:
A survey is given of results on the representation of solutions of systems of massless equations in terms of solutions of the Cauchy–Riemann equations on the space of Penrose twisters. A detailed introduction to the theory of twistors is given. The basic integral transform — the Penrose transformation — is investigated by means of complex integral geometry. Other approaches to the theory of twistors are discussed and compared. In addition, the Yang–Mills equation is considered from the point of view of nonlinear Cauchy–Riemann equations.
Citation:
S. G. Gindikin, G. M. Henkin, “The Penrose transform and complex integral geometry”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., 17, VINITI, Moscow, 1981, 57–111; J. Soviet Math., 21:4 (1983), 508–551
\Bibitem{GinHen81}
\by S.~G.~Gindikin, G.~M.~Henkin
\paper The Penrose transform and complex integral geometry
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat.
\yr 1981
\vol 17
\pages 57--111
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/intd48}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=628976}
\zmath{https://zbmath.org/?q=an:0507.53046|0482.53053}
\transl
\jour J. Soviet Math.
\yr 1983
\vol 21
\issue 4
\pages 508--551
\crossref{https://doi.org/10.1007/BF01084285}
Linking options:
https://www.mathnet.ru/eng/intd48
https://www.mathnet.ru/eng/intd/v17/p57
This publication is cited in the following 12 articles:
Qianqian Kang, Guangzhen Ren, Yun Shi, “The Radon–Penrose Transformation for Quaternionic k-Regular Functions on Right-Type Groups”, Adv. Appl. Clifford Algebras, 34:5 (2024)
B. Berndtsson, S. V. Kislyakov, R. G. Novikov, V. M. Polterovich, P. L. Polyakov, A. E. Tumanov, A. A. Shananin, C. L. Epstein, “Gennadi Markovich Henkin (obituary)”, Russian Math. Surveys, 72:3 (2017), 547–570
Qianqian Kang, Wei Wang, “On Penrose integral formula and series expansion of k-regular functions on the quaternionic space Hn”, Journal of Geometry and Physics, 64 (2013), 192
QianQian Kang, Wei Wang, “On Radon-Penrose transformation and k-Cauchy-Fueter operator”, Sci. China Math., 55:9 (2012), 1921
M. A. Akivis, V. V. Konnov, “Some local aspects of the theory of conformal structure”, Russian Math. Surveys, 48:1 (1993), 1–35
B. Yu. Sternin, V. E. Shatalov, “Differential equations on complex-analytic manifolds, and the Maslov canonical operator”, Russian Math. Surveys, 43:3 (1988), 117–148
M. M. Kapranov, Yu. I. Manin, “The twistor transformation and algebraic-geometric constructions of solutions of the equations of field theory”, Russian Math. Surveys, 41:5 (1986), 33–61
P. L. Polyakov, G. M. Henkin, “Homotopy formulas for the ¯∂-operator on CPn and the Radon–Penrose transform”, Math. USSR-Izv., 28:3 (1987), 555–587
S. G. Gindikin, “Integral formulas and integral geometry for ˉ∂-cohomologies in CPn”, Funct. Anal. Appl., 18:2 (1984), 108–119
Yu. I. Manin, I. B. Penkov, “Null-geodesics of complex Einstein spaces”, Funct. Anal. Appl., 16:1 (1982), 64–66
S. G. Gindikin, “Twistor representation of a non-self-dual metric”, Funct. Anal. Appl., 15:1 (1981), 55–57
S. G. Gindikin, “Twistor representation of solutions of a Maxwellian system”, Funct. Anal. Appl., 14:4 (1980), 298–300
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