Codazzi tensor

In the mathematical field of differential geometry, a Codazzi tensor (named after Delfino Codazzi) is a symmetric 2-tensor whose covariant derivative is also symmetric. Such tensors arise naturally in the study of Riemannian manifolds with harmonic curvature or harmonic Weyl tensor. In fact, existence of Codazzi tensors impose strict conditions on the curvature tensor of the manifold. Also, the second fundamental form of an immersed hypersurface in a space form (relative to a local choice of normal field) is a Codazzi tensor.
Definition.
Let formula_1 be a n-dimensional Riemannian manifold for formula_2, let formula_3 be a symmetric 2-tensor field, and let formula_4 be the Levi-Civita connection. We say that the tensor formula_3 is a Codazzi tensor if 
for all formula_7
Rigidity of Codazzi tensors.
Matsushima and Tanno showed that, on a Kähler manifold, any Codazzi tensor which is hermitian is parallel. Berger showed that, on a compact manifold of nonnegative sectional curvature, any Codazzi tensor with constant must be parallel. Furthermore, on a compact manifold of nonnegative sectional curvature, if the sectional curvature is strictly positive at least one point, then every symmetric parallel 2-tensor is a constant multiple of the metric.