In the mathematical field of differential geometry, a Codazzi tensor (named after Delfino Codazzi) is a symmetric 2-tensor whose covariant derivative is also symmetric. Such tensors arise 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. Also, the second fundamental form of an immersed hypersurface in a space form (relative to a local choice of normal field) is a Codazzi tensor.

Definition edit

Let   be a n-dimensional Riemannian manifold for  , let   be a symmetric 2-tensor field, and let   be the Levi-Civita connection. We say that the tensor   is a Codazzi tensor if

 

for all  

Examples edit

  • Any parallel (0,2)-tensor field is, trivially, Codazzi.
  • Let   be a space form, let   be a smooth manifold with   and let   be an immersion. If there is a global choice of unit normal vector field, then relative to this choice, the second fundamental form is a Codazzi tensor on   This is an immediate consequence of the Gauss-Codazzi equations.
  • Let   be a space form with constant curvature   Given any function   on   the tensor   is Codazzi. This is a consequence of the commutation formula for covariant differentiation.
  • Let   be a two-dimensional Riemannian manifold, and let   be the Gaussian curvature. Then   is a Codazzi tensor. This is a consequence of the commutation formula for covariant differentiation.
  • Let Rm denote the Riemann curvature tensor. Then div(Rm)=0 ("g has harmonic curvature tensor") if and only if the Ricci tensor is a Codazzi tensor. This is an immediate consequence of the contracted Bianchi identity.
  • Let W denote the Weyl curvature tensor. Then   ("g has harmonic Weyl tensor") if and only if the "Schouten tensor"
 
is a Codazzi tensor. This is an immediate consequence of the definition of the Weyl tensor and the contracted Bianchi identity.

Rigidity of Codazzi tensors edit

Matsushima and Tanno showed that, on a Kähler manifold, any Codazzi tensor which is hermitian is parallel. Berger showed that, on a compact manifold of nonnegative sectional curvature, any Codazzi tensor h with trgh constant must be parallel. Furthermore, on a compact manifold of nonnegative sectional curvature, if the sectional curvature is strictly positive at least one point, then every symmetric parallel 2-tensor is a constant multiple of the metric.

See also edit

References edit

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