Codazzi tensor: Difference between revisions

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Codazzi tensors (named after Delfino Codazzi) 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,Z)

.

References

Arthur Besse, Einstein Manifolds, Springer (1987).

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 ==Definition==
-Let <math>(M,g)</math> be a n-dimensional Riemannian manifold for <math>n \geq 3</math>, let <math>T</math> be a [[tensor]], and let <math>\nabla</math> be a [[Levi-Civita connection]] on the manifold. We say that the tensor <math>T</math> is a Codazzi Tensor if <math> (\nabla_X T) g(Y,Z) = (\nabla_Y T) g(X,Z) </math>.
+Let <math>(M,g)</math> be a n-dimensional Riemannian manifold for <math>n \geq 3</math>, let <math>T</math> be a [[tensor]], and let <math>\nabla</math> be a [[Levi-Civita connection]] on the manifold. We say that the tensor <math>T</math> is a Codazzi tensor if
+:<math> (\nabla_X T) g(Y,Z) = (\nabla_Y T) g(X,Z) </math>.
 ==See also==