a conformal differential invariant and the conformal rigidity of hypersurfaces

Authors

Maks A. Akivis, Ben-Gurion University of the Negev
Vladislav V. Goldberg, New Jersey Institute of Technology

Document Type

Article

Publication Date

1-1-1997

Abstract

For a hypersurface V^n-1 of a conformal space, we introduce a conformai differential invariant I = h²/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 ≥ 4, and let V^n-1 and V^n-1 be two nonisotropic hypersurfaces without umbilical points in a conformai space Cⁿ or a pseudoconformal space Cⁿ of signature (p, q), p = n -q. Suppose that there is a one-to-one correspondence f : V^n-1 V̄^n-1 between points of these hypersurfaces, and in the corresponding points of V^n-1 andV̄^n-1 the following condition holds: Ī = f,I, where f : T(V^n-1) → T(V̄^n-1 ) is a mapping induced by the correspondence J. Then the hypersurfaces V^n-1 and V^n-1~ are conformally equivalent. © 1997 American Mathematical Society.

Identifier

21744443511 (Scopus)

Publication Title

Proceedings of the American Mathematical Society

External Full Text Location

https://doi.org/10.1090/s0002-9939-97-03828-8

ISSN

00029939

First Page

2415

Last Page

2424

Issue

8

Volume

125

Recommended Citation

Akivis, Maks A. and Goldberg, Vladislav V., "a conformal differential invariant and the conformal rigidity of hypersurfaces" (1997). Faculty Publications. 16842.
https://digitalcommons.njit.edu/fac_pubs/16842