# Projections, modules and connections for the noncommutative cylinder

## TL;DR

This paper explores the structure of projections, modules, and geometric properties of the noncommutative cylinder, revealing parallels with the noncommutative torus and providing explicit geometric connections and curvature calculations.

## Contribution

It introduces a detailed analysis of projections, modules, and geometric structures on the noncommutative cylinder, including explicit curvature and connection formulas.

## Key findings

- Countable nontrivial projections in the algebra
- Concrete representatives for each $K_0$ class
- Explicit Levi-Civita connection and Gaussian curvature

## Abstract

We initiate a study of projections and modules over a noncommutative cylinder, a simple example of a noncompact noncommutative manifold. Since its algebraic structure turns out to have many similarities with the noncommutative torus, one can develop several concepts in a close analogy with the latter. In particular, we exhibit a countable number of nontrivial projections in the algebra of the noncommutative cylinder itself, and show that they provide concrete representatives for each class in the corresponding $K_0$ group. We also construct a class of bimodules endowed with connections of constant curvature. Furthermore, with the noncommutative cylinder considered from the perspective of pseudo-Riemannian calculi, we derive an explicit expression for the Levi-Civita connection and compute the Gaussian curvature.