Some examples of higher ank manifolds of nonnegative curvature

Abstract

Recall that a metric on M is locally irreducible if the universal cover of M does not split isometrically as a product. In nonpositive (sectional) curvature and higher rank, all locally irreducible finite volume manifolds (with bounded curvature) are locally symmetric spaces [1], [8], [12]. This result uses the special properties of nonpositive curvature in an essential way. In fact, Heintze found examples of normally homogeneous nonsymmetric spaces of nonnegative curvature and higher rank [16]. In this note, we will obtain more examples of higher rank and nonnegarive curvature with some new features. Indeed, the whole point of this paper is to show that higher rank metrics in nonnegative curvature can be very complicated. One should compare our situation with the pinching theorems. There there is a duality between positive and negative curvature. In fact, if M is any rank 1 compact locally symmetric space with nonconstant curvature then any other 1/4-pinched metric on M must be symmetric. For positive curvature, this is a consequence of Berger's famous rigidity theorem [ I0]. For negative curvature, this was proved by Hamenstfidt [ 15]. Notice though that there really is no theorem dual to the sphere theorem in negative curvature, due to the Gromov-Thurs ton examples of compact manifolds with arbitrarily pinched sectional curvatures which are not homotopy equivalent to a space with constant curvature. Similarly, our examples show that duality fails for the higher-rank rigidity theorems.