Elliptic spaces in Grassmann manifolds

Abstract

JOSEeH A. WoL 1. Introduction Let G,(F) denote the Grassmann manifold of n-dimensional subspaces of F , with its usual structure as a Riemannian symmetric space, where F denotes the real numbers, the complex numbers, or the quaternions. In an earlier paper [5] we studied the connected totally geodesic submunifolds B of G.(F) with the property that any two distinct elements of B have zero intersection as subspaces of F. We proved [5, Theorem 4] that B is iso- metric to a sphere, to a real, complex, or quaternionic projective space, or to the Cayley projective plane; we then [5, Theorem 8] classified (up to an isom- etry of Gn,(')) the manifolds B which are isometric to spheres. In Chapter I of this paper we show that B cannot be the Cayley projective plane (Theorem 2), and we classify the manifolds B which are not isometric to spheres (Theorem 3). The main technique is the application of the results of the preceding paper