Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.