[Paper Review] On Rigidly Scalar-Flat Manifolds

Boris Botvinnik, Brett McInnes|ArXiv.org|Nov 3, 1999

Geometry and complex manifolds22 references3 citations

TL;DR

This paper investigates compact, spin manifolds with zero scalar curvature that do not admit metrics of positive scalar curvature—termed 'rigidly scalar-flat' manifolds. Using special holonomy and index theory, the authors prove that such manifolds either have finite cyclic fundamental group or contradict the Gromov-Lawson-Rosenberg conjecture, providing a structural classification under spin and dimension constraints (dim ≥ 5).

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.

Motivation & Objective

• To understand the geometric and topological structure of compact manifolds that are scalar-flat but do not admit positive scalar curvature metrics.
• To explore the implications of zero scalar curvature in the context of generalized AdS/CFT correspondence, particularly when boundary scalar curvature vanishes.
• To classify rigidly scalar-flat manifolds using special holonomy and fundamental group properties.
• To investigate the relationship between scalar-flatness and the Gromov-Lawson-Rosenberg conjecture in the spin setting.
• To determine whether such manifolds can serve as counterexamples to known conjectures in scalar curvature geometry.

Proposed method

• Analyzes compact Riemannian manifolds with zero scalar curvature and no positive scalar curvature metric, focusing on spin structures.
• Applies techniques from special holonomy geometry, particularly the classification of manifolds with reduced holonomy groups.
• Uses index theory and the Dirac operator on spin manifolds to derive topological obstructions.
• Applies the Gromov-Lawson-Rosenberg conjecture as a framework to test the existence of positive scalar curvature metrics.
• Employs cohomological and fundamental group arguments to classify possible topological types of rigidly scalar-flat manifolds.
• Considers dimension constraints (dim ≥ 5) to ensure applicability of known results in spin geometry and scalar curvature obstructions.

Experimental results

Research questions

• RQ1 What topological and geometric constraints arise for compact spin manifolds that are scalar-flat but do not admit positive scalar curvature metrics?
• RQ2 How does the structure of the fundamental group relate to scalar-flatness and the absence of positive scalar curvature metrics?
• RQ3 Can rigidly scalar-flat manifolds be classified via their holonomy groups, particularly in the context of special holonomy?
• RQ4 Under what conditions does a rigidly scalar-flat manifold contradict the Gromov-Lawson-Rosenberg conjecture?
• RQ5 What implications does zero scalar curvature on the boundary have for generalized AdS/CFT correspondence in theoretical physics?

Key findings

• Any compact, spin manifold of dimension at least 5 that is rigidly scalar-flat must either have a finite cyclic fundamental group or be a counterexample to the Gromov-Lawson-Rosenberg conjecture.
• The absence of positive scalar curvature metrics on such manifolds is obstructed by topological invariants derived from index theory and spin structures.
• Special holonomy plays a key role in classifying the possible geometric types of rigidly scalar-flat manifolds.
• The results provide a structural dichotomy: either the fundamental group is finite cyclic, or the manifold violates the Gromov-Lawson-Rosenberg conjecture.
• The analysis confirms that scalar-flatness in the absence of positive scalar curvature metrics imposes strong topological restrictions in the spin setting.
• The findings extend the understanding of scalar curvature obstructions in higher-dimensional Riemannian geometry and have implications for theoretical physics, particularly in AdS/CFT contexts.