  In this paper, we prove a general maximum principle for the time dependent
Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on
complete Riemannian manifolds. As an application we construct complete
manifolds with bounded nonnegative sectional curvature of dimension greater
than or equal to four such that the Ricci flow does not preserve the
nonnegativity of the sectional curvature, even though the nonnegativity of the
sectional curvature was proved to be preserved by Hamilton in dimension three.
The example is the first of this type. This fact is proved through a general
splitting theorem on the complete family of metrics with nonnegative sectional
curvature, deformed by the Ricci flow.
