Turkish Journal of Mathematics
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 $M(n+1)$.
DOI
10.55730/1300-0098.3264
Keywords
Hypersurface, Bonnet's theorem, differential invariant
First Page
2208
Last Page
2230
Recommended Citation
SAĞIROĞLU, YASEMİN and GÖZÜTOK, UĞUR
(2022)
"Global differential invariants of nondegenerate hypersurfaces,"
Turkish Journal of Mathematics: Vol. 46:
No.
6, Article 12.
https://doi.org/10.55730/1300-0098.3264
Available at:
https://journals.tubitak.gov.tr/math/vol46/iss6/12