  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.
