The curvature tensor is the most important isometry invariant of a Riemannian metric. We study several related conditions on the curvature tensor to obtain topological and geo- metrical restrictions. The first condition is the that the kernel of the curvature tensor has codimension either two or three. In which case, we conclude that positive curvature can only occur on topologically trivial manifolds (for arbitrary dimension when the kernel is codimension two and only in dimension 4 for codimension three kernel). In the last half, we study the three dimensional manifolds with constant Ricci eigenvalues (λ, λ, 0). We obtain new examples of these, show that the fundamental group is free under basic assumptions, and give more explicit descriptions of the general case of these metrics.