Twisted Quadrics and Algebraic Submanifolds in ℝ n $\mathbb {R}^{n}$

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 ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. 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 ⋆-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 dS2, AdS2].