Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds

Josef Mikeš,Irina Tsyganok,Vladimir Rovenski,Sergey Stepanov

doi:10.3390/math9090927

Abstract

In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.

Highlights

Since the 1970s, complete Riemannian manifolds have been included in the range of research carried out using the Bochner technique
We discuss the global aspects of the geometry of locally conformally flat complete and compact
We summarize and improve some known results on the geometry in the large of locally conformally Riemannian manifolds

Summary

Introduction and Main Results

One of key analytical methods for differential geometry is the celebrated Bochner technique, which was founded by S. The main result of [6] is that an n-dimensional (n ≥ 3) connected complete locally conformally flat Riemannian manifold with positive constant scalar curvature is compact if. The main result of [19] states that if a complete locally conformally flat manifold of dimension n ≥ 3, whose Ricci tensor satisfies the inequality Ric ≥ 0, belongs to one of the following classes: either flat or locally isometric to the product of a canonical sphere and a line, the manifold is globally conformally equivalent to either Rn or a spherical space form (see [24]). A complete noncompact connected locally conformally flat Riemannian manifold of dimension n ≥ 4 with non-negative sectional curvature and constant scalar curvature is a flat space form, if the following inequality holds for some p ≥ 1:.

Proofs of Statements

Final Remarks