On Rigidly Scalar-Flat Manifolds

TL;DR

This paper investigates the properties of rigidly scalar-flat manifolds, especially those that are spin and of dimension five or higher, revealing their fundamental group characteristics and connections to the Gromov-Lawson-Rosenberg conjecture.

Contribution

It introduces the class of rigidly scalar-flat manifolds and analyzes their structure using special holonomy, linking their properties to fundamental group finiteness and the Gromov-Lawson-Rosenberg conjecture.

Findings

Rigidly scalar-flat manifolds with spin structure and dimension ≥ 5 have either finite cyclic fundamental groups or counterexamples to a major conjecture.

The study connects scalar flatness and special holonomy to topological and geometric properties.

Provides new insights into the structure of scalar flat manifolds in relation to boundary curvature conditions.

Abstract

Witten and Yau (hep-th/9910245) have recently considered a generalisation of the AdS/CFT correspondence, and have shown that the relevant manifolds have certain physically desirable properties when the scalar curvature of the boundary is positive. It is natural to ask whether similar results hold when the scalar curvature is zero. With this motivation, we study compact scalar flat manifolds which do not accept a positive scalar curvature metric. We call these manifolds rigidly scalar-flat. We study this class of manifolds in terms of special holonomy groups. In particular, we prove that if, in addition, a rigidly scalar flat manifold $M$ is $Spin$ with $\dim M\geq 5$, then $M$ either has a finite cyclic fundamental group, or it must be a counter example to Gromov-Lawson-Rosenberg conjecture.