A conformal differential invariant and the conformal rigidity of hypersurfaces

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.