Nonnegative curvature and conullity of the curvature tensor

The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity 3 in dimension four. We show that these conditions are compatible with nonnegative sectional curvature only if either the manifold is diffeomorphic to \(\mathbb {R}^n\) or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.