Codazzi tensor

Codazzi tensors arise very 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.

Definition

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 $(\nabla _{X}T)g(Y,Z)=(\nabla _{Y}T)g(X,Y)$.

Importance

Existence of Codazzi tensors impose strict conditions on the Curvature tensor of the manifold.

See Also

• Weyl-Schouten theorem

References

• Arthur Besse, Einstein Manifolds, Springer (1987).