Riemannian manifolds of dimension 7 whose skew-symmetric curvature operator has constant eigenvalues

Abstract

A Riemannian manifold is called IP, if the eigenvalues of its skew-symmetric curvature operator are pointwise constant. It was previously shown that for all n\ge 4, except n=7, any IP manifold either has constant curvature, or is a warped product, with some specific function, of a line and a space of constant curvature. We extend this result to the case n = 7, and also study 3-dimensional IP manifolds.