Twisted geometry for submanifolds of $\mathbb{R}^n$

TL;DR

This paper introduces a method for constructing noncommutative deformations of submanifolds in Euclidean space using Drinfel'd twist, extending classical differential geometry into a noncommutative setting.

Contribution

It presents a general procedure for applying Drinfel'd twist deformations to embedded submanifolds defined by smooth equations, advancing noncommutative geometry techniques.

Findings

Develops a systematic approach for noncommutative deformation of submanifolds.

Extends classical differential geometry to noncommutative frameworks.

Provides a foundation for further research in noncommutative geometry applications.

Abstract

This is a friendly introduction to our recent general procedure for constructing 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 pioneered in [Aschieri et al., Class. Quantum Gravity 23 (2006), 1883]; the commutative pointwise product is replaced by a (generally noncommutative) $\star$-product induced by a Drinfel'd twist.