  A decomposition theorem is established for a class of closed Riemannian
submanifolds immersed in a space form of constant sectional curvature. In
particular, it is shown that if $M$ has nonnegative sectional curvature and
admits a Codazzi tensor with "parallel mean curvature", then $M$ is locally
isometric to a direct product of irreducible factors determined by the spectrum
of that tensor. This decomposition is global when $M$ is simply connected, and
generalizes what is known for immersed submanifolds with parallel mean
curvature vector.
