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

DOI:
https://doi.org/10.1090/S0002-9939-97-03828-8

MathSciNet review:
1389501

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Abstract | References | Similar Articles | Additional Information

Abstract: For a hypersurface of a conformal space, we introduce a conformal differential invariant , where and are the first and the second fundamental forms of connected by the apolarity condition. This invariant is called the *conformal quadratic element* of . 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 . The main theorem states:

Let , and let and be two nonisotropic hypersurfaces without umbilical points in a conformal space or a pseudoconformal space of signature . Suppose that there is a one-to-one correspondence between points of these hypersurfaces, and in the corresponding points of and the following condition holds: where is a mapping induced by the correspondence . Then the hypersurfaces and are conformally equivalent.

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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

Email:
akivis@black.bgu.ac.il

**Vladislav V. Goldberg**

Affiliation:
Department of Mathematics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

Email:
vlgold@numerics.njit.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03828-8

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