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 23:1883, 2006); 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 \(\Xi _t\) of vector fields that are tangent to all the submanifolds that are level sets of the \(f^a\) (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted \(\Xi _t\). 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 \({\mathfrak {e}}\subset \Xi _t\). 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 \(\star \)-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 \(dS_2,AdS_2\)].

我们提出了一个通用程序来构造由一组平滑方程\(f^a(x)=0\)确定的\({\mathbb {R}}^n\)的嵌入子流形M的非交换变形。我们使用 Aschieri 等人的微分几何的 Drinfel'd 扭曲变形框架。(Class Quantum Gravity 23:1883, 2006)；交换的逐点乘积被替换为（通常是非交换的）\(\star \) -乘积由 Drinfel'd 扭曲确定。我们采用的扭曲基于向量场的李代数\(\Xi _t\)，这些向量场与作为\(f^a\) 的水平集的所有子流形相切（切线无穷小微分同胚）；扭曲嘉当微积分在扭曲\(\Xi _t\)下自动等变。我们可以始终如一地投射来自双绞线的连接\（{\ mathbb {R}} ^ N \）的扭曲中号，如果捻是基于合适的李代数\（{\ mathfrak {E}} \子集\僖_t \)。如果我们赋予\({\mathbb {R}}^n\)一个度量，那么扭曲和投影到法线和切线向量场交换，我们可以将 Levi-Civita 连接一致地投影到扭曲的M，前提是扭曲基于李子代数\({\mathfrak {k}}\subset {\mathfrak {e}}\)度量的 Killing 向量场；特别是一个扭曲的高斯定理。扭曲代数流形可以用生成器和\(\star \)多项式关系来表征。我们详细介绍了嵌入在扭曲欧几里得\({\mathbb {R}}^3\) 中的扭曲圆柱体和嵌入在扭曲 Minkowski \({\mathbb {R}}^3\) 中的扭曲双曲面[这些是扭曲的（反-)de Sitter 空间\(dS_2,AdS_2\) ]。