×

Nonrigid spherical real analytic hypersurfaces in \(\mathbb C^{2}\). (English) Zbl 1208.32031

Summary: A Levi nondegenerate real analytic hypersurface \(M\) of \(\mathbb C^{2}\) represented in local coordinates \((z,w) \in \mathbb C^{2}\) by a complex defining equation of the form \(w = \Theta (z,\bar z, \overline w)\), which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function \(\Theta \) satisfies an explicitly written sixth-order polynomial complex partial differential equation. In the rigid case (known before), this system simplifies considerably, but in the general nonrigid case, its combinatorial complexity shows well why the two fundamental curvature tensors constructed by E. Cartan [Ann. Sc. Norm. Super. Pisa, II. Ser. 1, 333–354 (1932; Zbl 0005.37401)] in his classification of hypersurfaces have, since then, never been reached in parametric representation.

MSC:

32V40 Real submanifolds in complex manifolds
32V05 CR structures, CR operators, and generalizations
32W50 Other partial differential equations of complex analysis in several variables

Citations:

Zbl 0005.37401
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Cartan É, Ann. Scuola Norm. Sup. Pisa 1 pp 333– (1932)
[2] DOI: 10.1088/0264-9381/20/23/004 · Zbl 1051.32019 · doi:10.1088/0264-9381/20/23/004
[3] DOI: 10.1016/j.geomphys.2004.09.003 · Zbl 1086.53023 · doi:10.1016/j.geomphys.2004.09.003
[4] DOI: 10.1080/17476930902759460 · Zbl 1163.32303 · doi:10.1080/17476930902759460
[5] DOI: 10.1155/S0161171201020117 · Zbl 1002.32029 · doi:10.1155/S0161171201020117
[6] Merker J, Bull. Soc. Math. France 129 pp 547– (2001)
[7] Merker J, Ann. Inst. Fourier (Grenoble) 52 pp 1443– (2002)
[8] DOI: 10.1007/s10958-005-0063-9 · doi:10.1007/s10958-005-0063-9
[9] Merker J, Ann. Fac. Sci. Toulouse pp 215– (2005)
[10] Segre B, Rend. Acc. Lincei, VI, Ser. 13 pp 676– (1931)
[11] Engel F, Unter Mitwirkung von Dr. Friedrich Engel, bearbeitet von Sophus Lie, B.G. Teubner, Leipzig, in: Theorie der Transformationsgruppen. Erster Abschnitt (1888)
[12] Lie S, Vorlesungen über Continuirlichen Gruppen, mit Geometrischen und Anderen Anwendungen (1893)
[13] Bryant RL, J. Élie Cartan 1998 & 1999, Inst. É. Cartan 16 pp 5– (2000)
[14] DOI: 10.1112/plms/s3-58.2.387 · Zbl 0675.58046 · doi:10.1112/plms/s3-58.2.387
[15] DOI: 10.1515/crll.1869.70.46 · JFM 02.0128.03 · doi:10.1515/crll.1869.70.46
[16] Cartan É, Bull. Soc. Math. France 52 pp 205– (1924)
[17] Merker J, Vanishing Hachtroudi curvature: An effective characterization of straightenability of contact pairs of Segre varieties
[18] Merker J, Int. Math. Res. Surveys 2006
[19] Lie S, in Gesammelte Abhandlungen 5 pp 240– (1924)
[20] Tresse A, Détermination des Invariants Ponctuels de l’Équation Différentielle du Second Ordre y”={\(\omega\)}(x, y, y’) (1896) · JFM 27.0254.01
[21] DOI: 10.1016/0022-0396(89)90154-X · Zbl 0671.34012 · doi:10.1016/0022-0396(89)90154-X
[22] DOI: 10.1007/s10440-006-9064-z · Zbl 1330.34016 · doi:10.1007/s10440-006-9064-z
[23] Merker J, Sophus Lie, Friedrich Engel et le problème de Riemann-Helmholtz (2010)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.