# Codazzi tensor

Let $(M,g)$ be a n-dimensional Riemannian manifold for $n \geq 3$, let $T$ be a tensor, and let $\nabla$ be a Levi-Civita connection on the manifold. We say that the tensor $T$ is a Codazzi Tensor if $(\nabla_X T) g(Y,Z) = (\nabla_Y T) g(X,Z)$.