Twisted submanifolds of R^n

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 Drinfeld 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 Drinfeld 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 $f^a$; 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 $mathfrak{e}subsetXi_t$. 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 $mathfrak{k}subsetmathfrak{e}$ 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 $dS_2,AdS_2$].