Cohomological Rigidity of Real Bott Manifolds

Abstract. A real Bott manifold is the total space of iterated RP 1 bundles starting with a point, where each RP 1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with 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.