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 Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} 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.