Non-existence of Riemannian metric satisfying Yamabe soliton

TL;DR

This paper proves that certain compact Riemannian manifolds cannot admit metrics with positive scalar curvature if they have a positive function satisfying specific Yamabe soliton conditions, revealing limitations on such geometric structures.

Contribution

It establishes the non-existence of Riemannian metrics with positive scalar curvature under specific Yamabe soliton conditions on compact manifolds.

Findings

No positive scalar curvature metric exists under the given conditions.

A relation between scalar curvature and surface area of geodesic balls is derived.

Results apply to manifolds with a pole satisfying steady Yamabe soliton.

Abstract

In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar curvature and surface area of a geodesic ball in a Riemannian manifold with a pole satisfying steady Yamabe soliton.