Search
Submit an article RU
Differential Geometry of Manifolds

Current issue

Back to the list Download the article
Stepanov S.E. Tsyganok I.I. Mikeš J.

Complete Riemannian manifolds with Killing — Ricci and Codazzi — Ricci tensors

DOI
10.5922/0321-4796-2022-53-10
Pages
112-117
complete Riemannian manifold Killing — Ricci tensor Codazzi — Ricci tensor Liouville-type theorems

Abstract

The purpose of this paper is to prove of Liouville type theorems, i. e., theorems on the non-existence of Killing — Ric­ci and Codazzi — Ricci tensors on complete non-com­pact Riemannian manifolds. Our results complement the two classical vanishing theorems from the last chapter of fa­mous Besse’s monograph on Einstein manifolds.

Reference

1. Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicata, 7:3, 259—280 (1978).

2. Suh, Y. J.: Generalized Killing Ricci tensor for real hypersurfaces in complex two-plane Grassmannians. J. Geom. Phys., 159, 103799 (2011). doi: 10.1016/j.geomphys.2020.103799.

3. Calvaruso, G.: Riemannian 3-metrics with a generic Codazzi Ricci tensor. Geom. Dedicata, 151:1, 259—267 (2011).

4. Petersen, P.: Riemannian Geometry. Springer (2016).

5. Besse, A. L.: Einstein Manifolds. Berlin Heidelberg (2008).

6. Stepanov, S. E.; Tsyganok, I. I.: Codazzi and Killing tensors on a complete Riemannian manifold. Math. Notes, 109:6, 901—911 (2021). (In Rus.).

7. Besse, A. L.: Einstein Manifolds. Springer (1987).

8. Bertrand, J., Sandeep, K.: Sharp Green’s function estimates on Ha­damard manifolds and Adams inequality. International Mathematics Re­search Notices, 6, 4729—4767 (2021).

9. Mikes, J., Rovenski, V., Stepanov, S.: On higher order Codazzi tensors on complete Riemannian manifolds. Annals of Global Analysis and Geometry, 56, 429—442 (2021).