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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.

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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