Codazzi tensor

Definition

Let (M,g) be a n-dimensional Riemannian manifold for n \geq 3, let T be a symmetric 2-tensor field, and let \nabla be the Levi-Civita connection. We say that the tensor T is a Codazzi tensor if : (\nabla_X T)(Y,Z) = (\nabla_Y T)(X,Z) for all X,Y,Z\in\mathfrak{X}(M).

Examples

• Any parallel (0,2)-tensor field is, trivially, Codazzi.
• Let (N,\overline{g}) be a space form, let M be a smooth manifold with 1+\dim M=\dim N, and let F:M\to N 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 M. This is an immediate consequence of the Gauss-Codazzi equations.
• Let (M,g) be a space form with constant curvature \kappa. Given any function f on M, the tensor \operatorname{Hess}^gf+\kappa fg is Codazzi. This is a consequence of the commutation formula for covariant differentiation.
• Let (M,g) be a two-dimensional Riemannian manifold, and let K be the Gaussian curvature. Then 2\operatorname{Hess}^gK+K^2g is a Codazzi tensor. This is a consequence of the commutation formula for covariant differentiation.
• Let Rm denote the Riemann curvature tensor. Then ("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 \operatorname{div}W=0 ("g has harmonic Weyl tensor") if and only if the "Schouten tensor" ::\operatorname{Ric}-\frac{1}{2n-2}Rg : is a Codazzi tensor. This is an immediate consequence of the definition of the Weyl tensor and the contracted Bianchi identity.

Rigidity

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 trg**h 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.

References

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