Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A conformal differential invariant
and the conformal rigidity of hypersurfaces

Authors: Maks A. Akivis and Vladislav V. Goldberg
Journal: Proc. Amer. Math. Soc. 125 (1997), 2415-2424
MSC (1991): Primary 53A30
MathSciNet review: 1389501
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Abstract: For a hypersurface $V^{n-1}$ of a conformal space, we introduce a conformal differential invariant $I = \frac {h^2}{g}$, where $g$ and $h$ are the first and the second fundamental forms of $V^{n-1}$ connected by the apolarity condition. This invariant is called the conformal quadratic element of $V^{n-1}$. The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of $V^{n-1}$. The main theorem states:

Let $n \geq 4$, and let $V^{n-1}$ and $\overline {V}^{n-1}$ be two nonisotropic hypersurfaces without umbilical points in a conformal space $C^n$ or a pseudoconformal space $C^n_q$ of signature $(p, q), \;\; p = n - q$. Suppose that there is a one-to-one correspondence $f: V^{n-1} \rightarrow \overline {V}^{n-1}$ between points of these hypersurfaces, and in the corresponding points of $V^{n-1}$ and $\overline {V}^{n-1}$ the following condition holds: $ \overline {I} = f_* I, $ where $f_*: T (V^{n-1}) \rightarrow T (\overline {V}^{n-1})$ is a mapping induced by the correspondence $f$. Then the hypersurfaces $V^{n-1}$ and $\overline {V}^{n-1}$ are conformally equivalent.

References [Enhancements On Off] (What's this?)

  • 1. M. A. Akivis and V. V. Goldberg, Projective differential geometry of submanifolds, North-Holland Mathematical Library, vol. 49, North-Holland Publishing Co., Amsterdam, 1993. MR 1234487
  • 2. Manfredo do Carmo and Marcos Dajczer, Conformal rigidity, Amer. J. Math. 109 (1987), no. 5, 963–985. MR 910359, 10.2307/2374496
  • 3. É. Cartan, Les sous-groupes des groupes continus de transformations. Ann. Sci. École Norm. (3) 25 (1908), 57-194. Jbuch. 39, pp. 206-207
  • 4. -, La déformation des hypersurfaces dans l'espace conforme réel à $n \geq 5$ dimensions. Bull. Soc. Math. France 45 (1917), 57-121. Jbuch. 46, p. 1129
  • 5. -, Sur la déformation projective des surfaces. Ann. Sci. École Norm. Sup. 37 (1920), 259-356. Jbuch. 47, pp. 656-657
  • 6. G. Fubini, Applicabilitá proiettiva di due superficie. Rend. Circ. Mat Palermo 41 (1916), 135-162. Jbuch. 46, pp. 1098-1099
  • 7. -, Studî relativi all' elemento lineare proiettivo di una ipersuperficie. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (5) 27 (1918), 99-106. Jbuch. 46, p. 1095
  • 8. G. Fubini and E. Cech, Geometria proiettiva differenziale. Zanichelli, Bologna, vol. 1, 1926, 394 pp., vol. 2, 1927, 400 pp. Jbuch. 52, pp. 751-752
  • 9. Robert B. Gardner, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1062197
  • 10. Gary R. Jensen and Emilio Musso, Rigidity of hypersurfaces in complex projective space, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 2, 227–248. MR 1266471
  • 11. Richard Sacksteder, The rigidity of hypersurfaces, J. Math. Mech. 11 (1962), 929–939. MR 0144286
  • 12. Christina Schiemangk and Rolf Sulanke, Submanifolds of the Möbius space, Math. Nachr. 96 (1980), 165–183. MR 600808, 10.1002/mana.19800960115
  • 13. Rolf Sulanke, Submanifolds of the Möbius space. III. The analogue of O. Bonnet’s theorem for hypersurfaces, Tensor (N.S.) 38 (1982), 311–317. MR 832662

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Additional Information

Maks A. Akivis
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Vladislav V. Goldberg
Affiliation: Department of Mathematics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

Keywords: Conformal and pseudoconformal geometry, hypersurface, first and second fundamental forms, conformal quadratic element, moving frames, conformal rigidity
Received by editor(s): November 21, 1995
Received by editor(s) in revised form: February 23, 1996
Additional Notes: This research was partially supported by Volkswagen-Stiftung (RiP-program at MFO). The research of the first author was also partially supported by the Israel Ministry of Absorption and the Israel Public Council for Soviet Jewry.
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society