One of the most interesting problems in Riemannian geometry is the study of the topology of manifolds which admit a metric with nonnegative sectional curvatures. Although many results are known, such as pinching theorems, in general the problem is quite open. In the case that the curvature operator is nonnegative, the results of several authors lead to a topological classification of such manifolds. This classification can be found in [MN]. If the dimension of the manifold is three, the nonnegativity of the sectional curvatures implies the nonnegativity of the curvature operator because the Weyl tensor is identically zero. If the dimension is four, the work of Walschap [W] gives a thorough understanding of complete noncompact 4-manifolds with nonnegative sectional curvatures.
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