论文标题
$ \ mathbb {r}^n $中的扭曲的四边形和代数submanifolds
Twisted Quadrics and Algebraic Submanifolds in $\mathbb{R}^n$
论文作者
论文摘要
我们提出了一个一般程序，以构建代数Submanifold $ m $ $ \ Mathbb {r}^n $的非交通变形，专门将过程[G. Fiore，T。Weber，$ \ mathbb {r}^n $，arxiv：2003.03854的扭曲子手机，对平滑的子曼属有效。我们使用Aschieri等人的扭曲微分几何形状的框架。 （Class。QuantumGrav。23，1883-1911，2006），通过$ \ star $ - 产品由Drinfel'd Twist确定的$ \ star $ - 产品代替。 We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra $Ξ_t$ of vector fields that are tangent to all the $M_c$.扭曲的cartan微积分自动在扭曲的$ξ_t$下自动进行。如果我们将$ \ mathbb {r}^n $赋予度量，然后将扭曲和投影到正常或切线组件通勤，将Levi-Civita连接到扭曲的$ M $一致，尤其是扭曲的高斯定理持有，只要扭曲基于杀死载体的扭曲。可以用发电机和$ \ star $ polynomial关系来表征扭曲的代数四边形。我们明确地根据$ \ m athbb {r}^3 $在所有四边形的曲折的情况下明确处理变形，但尤其是椭圆形的圆柱体，尤其是嵌入了扭曲的欧几里得$ \ mathbb {r}^r}^3 $ and twisted hampoloids的扭曲euclidean $ \ mathbb { [后者是扭曲的（抗）de Sitter Spaces $ DS_2 $，$ ADS_2 $]。
We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $\mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $\mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883-1911, 2006), whereby the commutative pointwise product is replaced by the $\star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra $Ξ_t$ of vector fields that are tangent to all the $M_c$. The twisted Cartan calculus is automatically equivariant under twisted $Ξ_t$. If we endow $\mathbb{R}^n$ with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted $M$ is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and $\star$-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb{R}^3$ except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $\mathbb{R}^3$ [the latter are twisted (anti-)de Sitter spaces $dS_2$, $AdS_2$].
