In this paper we investigate what kind of manifolds arise as the total spaces of iterated $S^1$-bundles. A real Bott tower studied in \cite{ CMO}, \cite{ KM} and \cite{ KN} is an example of an iterated $S^1$-bundle. We show that the total space of an iterated $S^1$-bundle is homeomorphic to an infra-nilmanifold. A real Bott manifold, which is the total space of a real Bott tower, provides an example of a closed flat Riemannian manifold. We also show that real Bott manifolds are the only closed flat Riemannian manifolds obtained from iterated $\bbr{ P}^1$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^1$-bundles in dimension 3.
Categories: math.AT math.DG math.GT