  We prove several Liouville-type non-existence theorems for higher order
Codazzi tensors and classical Codazzi tensors on complete and compact
Riemannian manifolds, in particular. These results will be obtained by using
theorems of the connections between the geometry of a complete smooth manifold
and the global behavior of its subharmonic functions. In conclusion, we show
applications of this method for global geometry of a complete locally
conformally flat Riemannian manifold with constant scalar curvature because its
Ricci tensor is a Codazzi tensor and for global geometry of a complete
hypersurface in a standard sphere because its second fundamental form is also a
Codazzi tensor.
