Higher order Codazzi tensors on conformally flat spaces
A symmetric \((0,m)\)-tensor field \(\phi\) is called a Codazzi tensor field with respect to an affine connection \(\nabla\) if and only if the covariant derivative of \(\phi\) with respect to \(\nabla \) is totally symmetric. Codazzi tensors naturally appear in the study of various classes of submanifolds. For example, parallel submanifolds of real and complex space forms are examples of manifolds admitting a Codazzi tensor of type \((0,2)\). Also, in affine differential geometry, the affine metric \(h\) can be seen as a Codazzi tensor of type \((0,2)\) with respect to the induced affine connection. In that setting, affine spheres can be characterized by the condition that the cubic form is a Codazzi tensor of type \((0,3)\) with respect to the Levi-Civita connection of the affine metric \(h\). In this paper, the authors give a short survey about the different domains in which such Codazzi tensors occur. In the case that the connection is the Levi-Civita connection of a semi-Riemannian metric \(g\), they also obtain a formula for the Laplacian of the length of \(\phi\). A formula, which under some additional assumptions (tracelessness of \(\phi\), conformally flat metric) becomes surprisingly nice. In the final section, some global consequences of this formula are presented, including some pinching theorems involving the Ricci curvature.
scientific article; zbMATH DE number 1208008