On space forms of Grassmann manifolds

Abstract

We show that, in each odd dimension n = m2, there is a class of Grassmann quotient spaces not included in Wolf's classic solution of the Grassmann space form problem. We classify all of these new Grassmann space forms up to isometry. As an application, we exhibit a pair of compact Einstein manifolds of dimension m2 with holonomy groups which are abstractly isomorphic yet not conjugate in the orthogonal group, thus proving that a theorem of Besse cannot be extended to non-simply-connected Einstein manifolds.