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.
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.
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 h with trgh 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.