On higher order Codazzi tensors on complete Riemannian manifolds

TL;DR

This paper establishes Liouville-type non-existence theorems for higher order and classical Codazzi tensors on complete and compact Riemannian manifolds, linking geometric properties with subharmonic function behavior.

Contribution

It introduces new Liouville-type theorems for Codazzi tensors and applies these to the global geometry of specific Riemannian manifolds and hypersurfaces.

Findings

Non-existence of certain Codazzi tensors under specified conditions

Applications to the geometry of conformally flat manifolds with constant scalar curvature

Insights into the geometry of hypersurfaces in spheres

Abstract

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.