Abstract
Necessary and sufficient conditions for the local integrability of systems of m linearly independent smooth complex vector fields defined on m +1 dimensional smooth real manifolds are given. Some similar results for the closely related abstract CR structures of hypersurface type are discussed also.
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References
A. V. Abrossimov, Complex differential systems and tangential Cauchy-Riemann equations, (in Russian), Math. Sbornik, 122 (164), no. 4(12), (1983), 419–434.
A. V. Abrossimov, On integrability of complex differential systems, (in Russian), Diff. Uravn. i ih Primen., Moscow, (1984), 131–138.
T. Akahori, A New Approach to the Local Embedding Theorem of CR-Structures for n ≥ 4, Memoirs of the AMS, no. 366, Providence, R.I., 1987.
A. Andreotti and C.D.Hill, Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Scuola Norm. Sup. Pisa, 26 Fasc. II (1972), 299–324.
M. S. Baouendi and L. P. Rothschild, Embeddability of abstract CR structures and integrability of related systems, Ann. Inst. Fourier, 37, no. 3, (1987), 131–141.
A. Boggess, CR Manifolds and Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, (1991).
L. Boutet de Monvel, Intégration des équations de Cauchy-Riemann induit formelles, Séminaire Goulaouic-Lions-Schwartz 1974–1975, exposé no. 9.
P. D. Cordaro and H.Jacobowitz (Editors), Multidimensional Complex Analysis and Partial Differential Equations, Proceedings of the Brazil-USA Conference on Multidimensional Complex Analysis and Partial Differential Equations, June 12–16, 1995, São Carlos, Brazil, Contemporary Mathemattics, no. 205, AMS, Providence, Rhode Island, 1997.
P. D.Cordaro and J. Hounie, Local solvability in C o ∞ of overdetermined systems of vector fields, J. of Func. Anal., 87, (1989), 231–249.
A. Djuraev, On the theory of overdetermined systems of first order (in Russian), Dokl. AN USSR, 321, no. 4, (1991), 664–669.
A. Djuraev, On some non-elliptic overdetermined systems of first order and their applications, (in Russian), Dokl. AN USSR, 321, no. 5, (1991), 885–891.
R. Dwilewicz, Embeddability of smooth Cauchy-Riemann manifolds, Ann. di Mat., 139, (1985), 15–43.
J. J. Faran, Non-analytic hypersurfaces in C n, Math. Ann., 226, (1977), 121–123.
S. J. Greenfield, Cauchy-Riemann equations in several variables, Ann. Sc. Normale Sup., 22 (1968), 275–314.
N. Hanges and H. Jacobowitz, A remark on almost complex structures with boundary, Amer. J. of Math., 111, (1989), 53–64.
G. M. Henkin, The Hans Lewy equation and analysis on pseudoconvex manifolds, I, (in Russian), Uspehi Math. Nauk, 32, no. 3 (195), (1977), 57–118.
L. Hörmander, An Introduction to Complex Analysis in Several Variables, D. Van Nostrand Company, Inc., Princeton, (1966).
H. Jacobowitz, An Introduction to CR Structures, Mathematical Surveys and Monographs, no. 32, AMS, (1990).
J. J. Kohn, Integration of complex vector fields, Bull. of the AMS, 78, no. 1, (1972), 1–11.
J. J. Kohn, Methods of partial differential equations in complex analysis, Several Complex Variables, Proc. of Symposia in Pure Math., AMS, vol. 30, (1977), 215–237.
M. Kuranishi, Strongly pseudoconvex CR structures over small balls, Ann. of Math., I, vol. 115, (1982), 451–500, II, vol. 116, (1982), 1–64, III, vol. 116, (1982), 249–330.
C. LeBrun, Twistor CR manifolds and three-dimensional conformal geometry, Trans. Amer. Math. Soc., vol. 284, no. 2, (1984), 601–616.
H. Lewy,On the local character of the solutions of an atypical linear differential equations in three variables and a related theorem for regular functions of two complex variables, Ann. of Math., 64, no. 3, (1956), 514–522.
H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math., 66, no. 1, (1957), 155–158.
G. A. Mendoza and F. Treves, Local solvability in a class of overdetermined systems of linear pde, Duke Math. J., 63, no. 2, (1991), 355–377.
A. Newlander and L. Nirenberg, Complex coordinates in almost complex manifolds, Annals of Math., (2) 65, (1957), pp.391–404.
L. Nirenberg, Lectures on Linear Partial Differential Equations, Reg. Conf. Series in Math., 17, (1973).
L. Nirenberg, On a question of Hans Lewy, Russian Math. Survey, 29, (1974), 251–262.
H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Math. Palermo, 23, (1907), 185–220, (or Oeuvres, IV, 224–289).
J. Tabov, Formally integrable complex linear systams of PDE’s with constant coefficients in R 3, C. R. Acad. bulg. Sci., 48, no. 9-10, (1995), 11-14; Formally integrable Mizohata systams in R 3, Nihonkai Math. J., 6, 2, (1995), 207–214.
G. Taiani, Cauchy-Riemann (CR) Manifolds, Math. Department, Pace University, New York 10038, (1989).
F. Treves, Approximation and Representation of Functions and Distributions Annihilated by a Systems of Complex Vector Fields, École Polytechnique, Centre de Mathématiques, (Mai 1981).
F. Treves, Homotopy Formulas in the Tangential Cauchy-Riemann Complex, Memoirs of the AMS, vol. 87, no. 434 (second of 3 numbers), (1990).
F. Treves, Hypo Analytic Structures, Princeton University Press, Princeton, New Jersey, (1992).
S. M. Webster, On the local solutions of the tangential Cauchy-Riemann equations, Ann. Inst. Henri Poincaré, 6, no. 3, (1989), 167–182.
S. M. Webster, On the proof of Kuranishi’s embedding theorem, Ann. Inst. Henri Poincaré, 6, no. 3, (1989), 183–207.
R. O. Wells, The Cauchy-Riemann equations and differential geometry, Bull. of AMS, 6, no. 2, (1982), 187–199.
K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds, Progress in Math., vol. 30, Birkhäuser, (1983).
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Apostolova, L.N. (2000). Local integrability of systems of m smooth linearly independent complex vector fields on m + 1 dimensional manifolds. In: de Arellano, E.R., Vasilevski, N.L., Shapiro, M., Tovar, L.M. (eds) Complex Analysis and Related Topics. Operator Theory Advances and Applications, vol 114. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8698-7_1
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DOI: https://doi.org/10.1007/978-3-0348-8698-7_1
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