The geometry of closed conformal vector fields on Riemannian spaces

TL;DR

This paper explores the geometric properties and obstructions related to closed conformal vector fields on Riemannian manifolds, extending classical theorems and providing new Bernstein-type results for hypersurfaces.

Contribution

It generalizes existing theorems on conformal vector fields, introduces obstructions on complete manifolds with nonpositive Ricci curvature, and extends Bernstein-type theorems to broader classes of hypersurfaces.

Findings

Obstructions to existence of certain conformal vector fields on complete manifolds.

Difficulty in finding non-trivial examples of spaces with multiple closed conformal vector fields.

Generalized Bernstein-type theorems for hypersurfaces in Riemannian manifolds with closed conformal vector fields.

Abstract

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.