论文标题
R^n的扭曲的子手机
Twisted submanifolds of R^n
论文作者
论文摘要
我们提出了一个一般过程，以构建嵌入式submanifold $ m $ $ \ mathbb {r}^n $由一组平滑方程式$ f^a（x）= 0 $确定的非交通变形。我们使用[Aschieri等，class的差异几何形状的扭曲变形的框架。量子重力23（2006），1883年]；换向点的产品被（通常是非交通性的）$ \ star $ - 产品取代，由drinfel twist确定。我们采用的曲折基于向量字段的lie代数$ξ_t$，这些vector字段与$ f^a $的所有级别集合的所有submanifolds相切；扭曲的cartan微积分自动在扭曲的切线无穷小差异下自动均等。如果扭曲是基于合适的谎言subalgebra $ \ mathfrak {e} \ subset接受了ξ_t$，我们可以始终将连接从扭曲的$ \ mathbb {r}^n $转换为扭曲的$ m $。 If we endow $\mathbb{R}^n$ with a metric then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi-Civita connection consistently to the twisted $M$, provided the twist is based on the Lie subalgebra $\mathfrak{k}\subset\mathfrak{e}$ of the Killing vector fields of the metric;尤其是扭曲的高斯定理。扭曲的代数歧管可以用发生器和多项式关系来表征。我们在一些细节中介绍嵌入在扭曲的欧几里德$ \ mathbb {r}^3 $的扭曲圆柱体和嵌入在扭曲的Minkowski $ \ Mathbb {r}^3 $中的扭曲的倍或扭曲的倍或倍增的倍增型倍增（这些是扭曲的（twisted de twisted（anti-）de tepte（anti-）de sitter Spaces $ ds_2 $ ds_2，ads_2 $]。
We propose a general procedure to construct noncommutative deformations of an embedded submanifold $M$ of $\mathbb{R}^n$ determined by a set of smooth equations $f^a(x)=0$. We use the framework of Drinfel'd twist deformation of differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006), 1883]; the commutative pointwise product is replaced by a (generally noncommutative) $\star$-product determined by a Drinfel'd twist. The twists we employ are based on the Lie algebra $Ξ_t$ of vector fields that are tangent to all the submanifolds that are level sets of the $f^a$; the twisted Cartan calculus is automatically equivariant under twisted tangent infinitesimal diffeomorphisms. We can consistently project a connection from the twisted $\mathbb{R}^n$ to the twisted $M$ if the twist is based on a suitable Lie subalgebra $\mathfrak{e}\subsetΞ_t$. If we endow $\mathbb{R}^n$ with a metric then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi-Civita connection consistently to the twisted $M$, provided the twist is based on the Lie subalgebra $\mathfrak{k}\subset\mathfrak{e}$ of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $\mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $\mathbb{R}^3$ [these are twisted (anti-)de Sitter spaces $dS_2,AdS_2$].
