Hypersurfaces with $L_r$-Pointwise 1-Type Gauss Map
DOI:
https://doi.org/10.15407/mag14.01.067Анотація
У статтi вивчаються гiперпрверхнi в $\mathbb{E}^{n+1}$, гауссове вiдображення $G$ яких задовольняє рiвняння $L_r G = f(G + C)$ для гладкої функцiї $f$ i постiйного вектора $C$, де $C$ є лiнеаризованим оператором $(r+1)$-ої середньої кривизни гiперповерхнi, тобто $L_r(f) = {\mathrm Tr}(P_r \circ \nabla^2f)$ для $f \in C^{\infty}(M)$, а $P_r$ є $r$-им перетворенням Ньютона, $\nabla^2 f$ є гессiаном $f$, $L_rG = (L_rG_1, \ldots ,L_rG_{n+1})$ і $G = (G_1,\ldots ,G_{n+1})$. Наша увага зосереджена на гiперповерхнях з постiйною $(r + 1)$-ою средньою кривизною i постiйною середньою кривизною. Для цих класiв гiперповерхонь отримано теореми класифiкацiї i характеризацiї.
Mathematical Subject Classification: 53D02, 53C40, 53C42
Ключові слова:
лiнеаризованi оператори $L_r$, $L_r$-точкове типу 1 гауссове вiдображення, $r$-мiнiмальна гiперповерхняПосилання
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