Global differential invariants of nondegenerate hypersurfaces

Abstract

Let $\{g_{ij}(x)\}_{i, j=1}^n$ and $\{L_{ij}(x)\}_{i, j=1}^n$ be the sets of all coefficients of the first and second fundamental forms of a hypersurface $x$ in $R^{n+1}$. For a connected open subset $U\subset R^{n}$ and a $C^{\infty }$-mapping $x:U\rightarrow R^{n+1}$ the hypersurface $x$ is said to be $d$-\textit{nondegenerate}, where $d\in \left\{1, 2, \ldots n\right\}$, if $L_{dd}(x)\neq 0$ for all $u\in U$. Let $M(n)=\{F:R^{n}\longrightarrow R^{n}\mid Fx=gx+b, \; g\in O(n), \; b\in R^{n}\}$, where $O(n)$ is the group of all real orthogonal $n\times n$-matrices, and $SM(n)=\{F\in M(n)\mid g\in SO(n)\}$, where $SO(n)=\left\{g\in O(n)\mid \det(g)=1\right\}$. In the present paper, it is proved that the set $\left\{g_{ij}(x), L_{dj}(x), i, j=1, 2,\ldots ,n\right\}$ is a complete system of a $SM(n+1)$-invariants of a $d$-non-degenerate hypersurface in $R^{n+1}$. A similar result has obtained for the group $M(n+1)$.