Codazzi tensor

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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[edit]

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) .

See also[edit]

References[edit]

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