Twisted submanifolds of Rn

We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of Rn determined by a set of smooth equations fa(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) ⋆ -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 fa (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted Ξ t. We can consistently project a connection from the twisted Rn to the twisted M if the twist is based on a suitable Lie subalgebra e⊂ Ξ t. If we endow Rn 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 k⊂ 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 R3 and twisted hyperboloids embedded in twisted Minkowski R3 [these are twisted (anti-)de Sitter spaces dS2, AdS2].