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{{citation style|date=June 2012}}
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{{context|date=June 2012}}
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 [[Riemann curvature tensor|Curvature tensor]] of the manifold.
 
==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>.
 
==See also==
*[[Weyl–Schouten theorem]]
 
==References==
* Arthur Besse, ''Einstein Manifolds'', Springer (1987).
 
[[Category:Tensors]]

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