Cohomological rigidity of real Bott manifolds

Abstract

A real Bott manifold is the total space of an iterated ${ℝ\mathbb{P}}^1$–bundle over a point, where each ${ℝ\mathbb{P}}^1$–bundle is the projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with $\mathbb{Z}∕2$–coefficients are isomorphic.

A real Bott manifold is a real toric manifold and admits a flat Riemannian metric invariant under the natural action of an elementary abelian 2–group. We also prove that the converse is true, namely a real toric manifold which admits a flat Riemannian metric invariant under the action of an elementary abelian 2–group is a real Bott manifold.