Codazzi Tensors and Reducible Submanifolds

An integral formula is derived for Codazzi tensors of type $(k, k)$. Many of the classical Minkowski type integral formulas then become special cases of this one. If $M$ is a submanifold of Euclidean space and $\pi$ is a parallel distribution on $M$ then each leaf of $\pi$ is a submanifold of Euclidean space with mean curvature normal vector field $\eta$. Using the above integral formula we show that the integral of ${\left | \eta \right |^2}$ over $M$ is bounded below by an intrinsic constant and we give necessary and sufficient conditions for equality to hold. The reducible surfaces for which equality holds are characterized and related results concerned with Riemannian product manifolds are proved. Parallel tensors of type $(1, 1)$ are characterized in terms of the de Rham decomposition. It is shown that if $M$ is irreducible and $A$ is a parallel tensor of type $(1, 1)$ on $M$ which is not multiplication by a constant then $M$ is a Kaehler manifold. Some further results are derived for manifolds whose simply connected cover is Kaehler.